Gain and phase imbalance compensation for OFDM systems

ABSTRACT

The present invention includes methods and devices to compensate for gain and phase imbalance for OFDM and other multi-carrier symbol transmission systems. More particularly, methods and devices for determining compensation parameters are provided. This invention may be applied to a variety of standards utilizing OFDM technology, including IEEE 802.11a, Hiperlan/2 and MMAC. Further description of the invention and its embodiments are found in the figures, specification and claims that follow.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is related to the commonly owned, concurrently filedU.S. patent application Ser. No. 10/259,108, entitled “Cubic SplinePredistortion, Algorithm and Training, for a Wireless LAN System” by thesame inventor. The related application is hereby incorporated byreference.

BACKGROUND OF THE INVENTION

The present invention includes methods and devices to compensate forgain and phase imbalance for OFDM and other multi-carrier symboltransmission systems. More particularly, methods and devices fordetermining compensation parameters are provided. This invention may beapplied to a variety of standards utilizing OFDM technology, includingIEEE 802.11a, Hiperlan/2 and MMAC.

Orthogonal frequency division multiplexing (OFDM) is a spectrallyefficient modulation scheme with application in both wired and wirelesscommunications. It is applied in existing wireless systems includingIEEE 802.11a and 802.11g and is proposed for several next-generationwireless systems including IEEE 802.16 and 4th generation cellular.Because OFDM has high spectral efficiency, it is more susceptible toradio impairments. One such impairment, and the topic of this study, isgain and phase imbalance (also known as IQ offset).

Gain and phase imbalance are introduced at both the transmit and receiveradios due to the typical variations in mixer components. The D/A andA/D converters may also introduce some gain imbalance; however, thisimbalance can be treated as combined with that of the mixer. Gainimbalance occurs whenever the gains in the in-phase and quadrature pathsare not identical and phase imbalance occurs whenever the phaseseparation of the in-phase and quadrature paths is not exactly 90degrees. Typical gain and phase imbalances of analog mixers are on theorder of 1 dB and 5 degrees, respectively. The effect of imbalance is aloss of orthogonality between the in-phase and quadrature paths. Thisloss of orthogonality can seriously degrade the link performance of anOFDM system, particularly when larger constellations are transmitted.

A major challenge in OFDM transceiver design is ensuring gain and phasebalance at the transmitter and the receiver. Standards for OFDMtransmission typically lack specific requirements for gain and phasebalance, relying instead on performance standards that necessitate goodbalance. Typically, designers focus on improving the mixer section ofthe system, to avoid imbalance. This comes at the cost of a relativelyexpensive mixer section.

Accordingly, an opportunity arises to develop gain and imbalancecompensation method and apparatus that improves system performance.

SUMMARY OF THE INVENTION

The present invention includes methods and devices to compensate forgain and phase imbalance for OFDM and other multi-carrier symboltransmission systems. More particularly, methods and devices fordetermining compensation parameters are provided. This invention may beapplied to a variety of standards utilizing OFDM technology, includingIEEE 802.11a, Hiperlan/2 and MMAC. Further description of the inventionand its embodiments are found in the figures, specification and claimsthat follow.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a generic OFDM transceiver model used in this study.

FIG. 2 illustrates the eye diagram of a 64 QAM system with this typicalimbalance applied only at the transmitter.

FIG. 3 shows a similar figure but with both the transmitter and receiversuffering 1 dB gain imbalance and 5° phase offset.

FIG. 4 illustrates the signal-to-distortion ratio for either gain orphase imbalance applied, individually, at either the transmitter orreceiver.

DETAILED DESCRIPTION

The following detailed description is made with reference to thefigures. Preferred embodiments are described to illustrate the presentinvention, not to limit its scope, which is defined by the claims. Thoseof ordinary skill in the art will recognize a variety of equivalentvariations on the description that follows.

This describes an OFDM symbol transmission and receiver model,particularly an 802.11a system model, and analyzes the effects of gainand phase imbalance. A method is proposed for measuring the imbalance ata receiver assuming OFDM modulation. Once the imbalance is measured, itis compensated for. Optionally, compensation can be performed at boththe transmitter and receiver. Transmitter based compensationpre-compensates for imbalance introduced at the transmit mixer. Tworeceiver based compensation algorithms are defined: one corrects forimbalance introduced at the receiver and the other compensates forimbalance introduced at the far-end transmitter. Although the primarygoal is improving link performance, imbalance compensation will increaseproduction yields by reducing the transceiver's sensitivity to mixerimbalance.

An alternate solution (besides specifying a high performance, and thusmore expensive mixer) is to perform mixing in the digital domain. Thedisadvantage of digital mixing is the high sampling rate D/A's and A/D'sthat would be required (potentially equaling the costs of more expensivemixers). Furthermore, even if a digital mixer is used at the receiver,there is no guarantee that the far-end transmit mixer will be digital(or even a well balanced analog mixer).

GLOSSARY AND NOMENCLATURE

Many symbols are used in this application, so their meanings are set outhere for reference.

-   -   A/D Analog to Digital Converter    -   D/A Digital to Analog Converter    -   dB Decibel    -   FFT Fast Fourier Transform    -   IEEE Institute of Electrical and Electronic Engineers    -   IFFT Inverse Fast Fourier Transform    -   IQ In-phase/Quadrature    -   ISI Inter-Symbol Interference    -   LAN Local Area Network    -   LNA Low Noise Amplifier    -   OFDM Orthogonal Frequency Division Multiplexing    -   PA Power Amplifier    -   RX Receiver    -   TX Transmitter

System Model

FIG. 1 illustrates a generic OFDM transceiver model used in this study.It is not comprehensive, but rather highlights those system componentsrequired to analyze the effects of imbalance and the correspondingcompensation schemes. Imbalance compensation is shown in both thetransmit and receive chains for both frequency domain and time domainapproaches. In practice, the transmitter typically will have either timeor frequency domain TX imbalance compensation and the receiver will havefrequency domain TX imbalance compensation and either time or frequencydomain RX imbalance compensation. OFDM transceiver standards typicallydo not mention or mandate imbalance compensation. Typical performancerequirements make it desirable to use either well-balanced mixers orimbalance compensation to ensure compliance.

The FIG. 1 model incorporates all the functionality of the 802.11atransmitter and receiver at the OFDM symbol level and below. It does notincorporate the 802.11a framing, scrambling, or convolutionalencoding/interleaving, which were not necessary for this evaluation. Theradio model consists of digital-to-analog converters (D/A's) 109, 111, amixer 113, 115, 117 and a PA 117. A predistorter may be inserted betweenthe OFDM transmitter and radio, as part of a time domain compensator 101or a frequency domain compensator 107 to pre-compensate fornon-linearities introduced by the PA. The predistorter is not mandated(or mentioned) in the 802.11a standard. IEEE Std 802.11a Part 11:Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY)Specifications: High-speed Physical Layer in the 5 GHz Band, 1999.Reconstruction filters between the D/A's and the PA are assumed to havenegligible effects and thus are ignored in the analysis. Those of skillin the art will recognize that additional steps may be implemented toaddress ripple and delay introduced by reconstruction filters. The timedomain 101 and frequency domain 107 compensators also may be adapted tocompensate for gain and phase imbalances.

OFDM symbols typically include 48 data carriers and 4 pilot carriers.The data carriers are modulated by a set of data symbols whose valuesfor the purpose of this evaluation are independent, uniformlydistributed random variables over the range of 0, 1, . . . , 2^(N)^(bpsc) −1 where N_(bpsc) represents the number of bits encoded ontoeach subcarrier. N_(bpsc) may equal to 1, 2, 4 and 6 bits correspondingto gray-coded BSPK, QPSK, 16-QAM and 64-QAM constellations,respectively. The QAM Mapper maps each data symbol to its correspondingcomplex value. The pilot carriers are encoded as BPSK symbols withpseudo-randomly alternating phases in accordance with 802.11a.

The data and pilot carriers are combined into d⁽¹⁾ _(n), n=0, . . . , 63complex points, and translated to the time domain by an N=64-pointcomplex IFFT 103. Each block of N samples output from the IFFT forms asingle OFDM symbol, which is then cyclic extended (105) with N_(cp)=16samples and windowed 105. Although 802.11a specifies a raised cosinewindow of arbitrary length w_(T) (where w_(T) represents the number ofsamples overlap between adjacent OFDM symbols), its use is optional. Forthis analysis, windowing with w_(T)=1 was used to meet the PSDrequirement of 802.11a. Oversampling of the transmit signal can beperformed either in the time-domain using standard interpolationtechniques or in the frequency domain by performing an osr×N-point IFFTon a zero padded input, where osr represents the oversampling rate. Forthis analysis, oversampling is performed in the frequency domain andN_(cp) and w_(T) are correspondingly scaled by osr.

The two D/A's convert the in-phase 107 and quadrature 109 components ofthe digital baseband signal 113, 115 to the analog domain for input tothe mixer. The mixer 117 is modeled as a direct-to-RF mixer, for thisanalysis. A power amplifier 119 boosts the signal for transmissionthrough a channel 121, which typically injects noise 123.

On the receiver side, the model components generally correspond to thetransmitter components. A low noise amplifier (LNA) 149 is used to boostthe received signal for processing. The in phase 143 and 145 quadraturecomponents of the signal are reconstructed. A pair of A/D's convert thecomponents from the analog domain to the digital domain. A receiverimbalance compensator 137 may be applied to correct for gain and phaseimbalance, in the time domain. A sync component 135 processes the signalbefore a Fast Fourier Transform 133 is applied to convert the signalfrom the time to frequency domain. An equalizer 132 uses one or morereference carriers to equalize the received signal. Additionalcompensation may be applied in the frequency domain 131.

In this analysis, the following simplifying assumptions are made: Theduration of the channel impulse response is shorter than the duration ofthe cyclic extension minus the duration of the windowing overlap. Thisallows the cyclic extension and any windowing operations to be omittedfrom this analysis. And, the carrier frequency offset betweenfrequencies of the transmitter and receiver is assumed to be negligible.Those of skill in the art will recognize that the equations presentedhere may be generalized to include such offsets.

End-to-end frequency-domain models describing transmitter and receiverimbalance are derived here for both no channel and multipath channelcases. These frequency domain models relate the received complexconstellation points to the transmitted points. From this, methods ofmeasuring and compensating for imbalance are developed.

No Channel

This simplified model considers the impact of transmitter and receiverimbalance when there is no channel. This exercise is useful forcharacterizing the impact of imbalance (without other impairments), aswell as for determining how to compensate for TX imbalance at thetransmitter since TX compensation at the transmitter is independent ofthe channel.

Consider the set of N complex constellation points a_(k)=a^(R)_(k)+ja^(I) _(k), k=0, 1, . . . , N−1 which are transmitted as a singleOFDM symbol. These are converted from the frequency domain to the timedomain using an N-point IFFT as follows

$\begin{matrix}{{x(n)} = {\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}{\left( {a_{k}^{R} + {j\; a_{k}^{I}}} \right)\left\lbrack {{\cos\left( {2\;\pi\;{{nk}/N}} \right)} + {j\;{\sin\left( {2\;\pi\;{{nk}/N}} \right)}}} \right\rbrack}}}} \\{= {{\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}\left\lbrack {{a_{k}^{R}{\cos\left( {2\;\pi\;{{nk}/N}} \right)}} - {a_{k}^{I}{\sin\left( {2\;\pi\;{{nk}/N}} \right)}}} \right\rbrack}} +}} \\{j\;\frac{1}{N}{\sum\limits_{k = 0}^{N - 1}\left\lbrack {{a_{k}^{R}{\sin\left( {2\;\pi\;{{nk}/N}} \right)}} + {a_{k}^{I}{\cos\left( {2\;\pi\;{{nk}/N}} \right)}}} \right\rbrack}}\end{matrix}$for n=0, 1, . . . , N−1. The IFFT output, x(n), is passed through thetransmitter's TX I/Q Imbalance Compensator to obtain y(n). For thedevelopment of this model, the imbalance compensator is disabled suchthat y(n)=x(n). The in-phase and quadrature components of y(n) are eachpassed through a D/A converter and then mixed to some carrier frequencyf_(c) to obtain the real valued RF signalz(t)=y _(I)(t)cos(2πf _(c) t)−y _(Q)(t)G ^(tx) _(m)sin(2πf _(c) t+θ^(tx) _(m))where G^(tx) _(m) and θ^(tx) _(m) represent the gain and phase imbalanceof the transmit mixer, D/A's and amplifiers referenced to the in-phasecomponent. Assuming no channel impairments, the receiver mixes thereceived RF signal back to complex baseband (filtering out the imagefrequencies and scaling by a factor of 2)

$\begin{matrix}{{{\hat{y}}_{I}(t)} = {2\left\{ {\left\lbrack {{{y_{I}(t)}{\cos\left( {2\pi\; f_{c}t} \right)}} - {{y_{Q}(t)}G_{m}^{tx}{\sin\left( {{2\pi\; f_{c}t} + \theta_{m}^{tx}} \right)}}} \right\rbrack{\cos\left( {2\pi\; f_{c}t} \right)}} \right\}_{LPF}}} \\{= {{y_{I}(t)} - {{y_{Q}(t)}G_{m}^{tx}{\sin\left( \theta_{m}^{tx} \right)}}}} \\{{{\hat{y}}_{Q}(t)} = {2\left\{ {{- \left\lbrack {{{y_{I}(t)}{\cos\left( {2\pi\; f_{c}t} \right)}} - {{y_{Q}(t)}G_{m}^{tx}{\sin\left( {{2\pi\; f_{c}t} + \theta_{m}^{tx}} \right)}}} \right\rbrack}G_{m}^{rx}{\sin\left( {{2\pi\; f_{c}t} +} \right.}} \right.}} \\\left. \left. \theta_{m}^{rx} \right) \right\}_{LPF} \\{= {G_{m}^{rx}\left\lbrack {{{y_{Q}(t)}G_{m}^{tx}{\cos\left( {\theta_{m}^{tx}\; - 0_{m}^{rx}} \right)}} - {{y_{I}(t)}{\sin\left( \theta_{m}^{rx} \right)}}} \right\rbrack}}\end{matrix}$where G^(rx) _(m) and θ^(rx) _(m) represent the combined gain and phaseimbalance of the receive mixer, A/D's and amplifiers, again referencedto the in-phase component. Converting to the digital domain andrewriting the received baseband signal in complex notationŷ(n)=y _(I)(n)[1−jG ^(rx) _(m)sin(θ^(rx) _(m))]+jy _(Q)(n)[G ^(rx) _(m)G ^(tx) _(m)cos(θ^(tx) _(m)−θ^(rx) _(m))+jG ^(tx) _(m)sin(θ^(tx) _(m))].

Defined two time-invariant intermediary valuesα=1−jG ^(rx) _(m)sin(θ^(rx) _(m))β=G ^(rx) _(m) G ^(tx) _(m)cos(θ^(tx) _(m)−θ^(rx) _(m))+jG ^(tx)_(m)sin(θ^(tx) _(m))such thatŷ(n)=αy _(I)(n)+jβy _(Q)(n).Ignoring the effects of the receiver's time-domain I/Q imbalancecompensator, {circumflex over (x)}(n)=ŷ(n). The reference to whetherthis compensates for transmitter or receiver imbalance is intentionallyomitted because, for the case of no channel, both transmitter andreceiver imbalance can be compensated for in the time domain. Thereceived set of constellation points is decoded using the FFT as follows

$\begin{matrix}{{\hat{a}}_{k} = {\sum\limits_{n = 0}^{N - 1}{\left\lbrack {{\alpha\;{y_{I}(n)}} + {j\;\beta\;{y_{Q}(n)}}} \right\rbrack\left\lbrack {{\cos\left( {2\;\pi\;{{nk}/N}} \right)} - {j\;{\sin\left( {2\;\pi\;{{nk}/N}} \right)}}} \right\rbrack}}} \\{= {\frac{1}{N}{\sum\limits_{n = 0}^{N - 1}{\begin{bmatrix}{{\alpha\;{\sum\limits_{k^{\prime} = 0}^{N - 1}\left\lbrack {{a_{k^{\prime}}^{R}{\cos\left( {2\;\pi\;{{nk}^{\prime}/N}} \right)}} - {a_{k^{\prime}}^{I}{\sin\left( {2\;\pi\;{{nk}^{\prime}/N}} \right)}}} \right\rbrack}} +} \\{j\;\beta{\sum\limits_{k^{\prime} = 0}^{N - 1}\left\lbrack {{a_{k^{\prime}}^{R}{\sin\left( {2\;\pi\;{{nk}^{\prime}/N}} \right)}} + {a_{k^{\prime}}^{I}{\cos\left( {2\;\pi\;{{nk}^{\prime}/N}} \right)}}} \right\rbrack}}\end{bmatrix}\left\lbrack {{\cos\left( {2\;\pi\;{{nk}/N}} \right)} - {j\;{\sin\left( {2\;\pi\;{{nk}/N}} \right)}}} \right\rbrack}}}} \\{= {\frac{1}{N}{\sum\limits_{n = 0}^{N - 1}{\sum\limits_{k^{\prime} = 0}^{N - 1}{\begin{bmatrix}{{a_{k^{\prime}}^{R}\left\lbrack {{\alpha\;{\cos\left( {2\;\pi\;{{nk}^{\prime}/N}} \right)}} + {j\;\beta\;{\sin\left( {2\;\pi\;{{nk}^{\prime}/N}} \right)}}} \right\rbrack} +} \\{j\;{a_{k^{\prime}}^{I}\left\lbrack {{\beta\;{\cos\left( {2\;\pi\;{{nk}^{\prime}/N}} \right)}} + {j\;\alpha\;{\sin\left( {2\;\pi\;{{nk}^{\prime}/N}} \right)}}} \right\rbrack}}\end{bmatrix}\left\lbrack {{\cos\left( {2\;\pi\;{{nk}/N}} \right)} - {j\;{\sin\left( {2\;\pi\;{{nk}/N}} \right)}}} \right\rbrack}}}}}\end{matrix}$for k=0, 1, . . . , N−1. This can be simplified into the following form(where, for convenience,A=2πnk′/NandB=2πnk/N)

$\begin{matrix}{{\hat{a}}_{k} = {\frac{1}{2N}{\sum\limits_{n = 0}^{N - 1}{\sum\limits_{k^{\prime} = 0}^{N - 1}\begin{bmatrix}{{\left( {{\alpha\; a_{k^{\prime}}^{R}} + {j\;\beta\; a_{k^{\prime}}^{I}} - {\beta\; a_{k^{\prime}}^{R}} - {j\;\alpha\; a_{k^{\prime}}^{I}}} \right){\cos\left( {A + B} \right)}} +} \\{{\left( {{\alpha\; a_{k^{\prime}}^{R}} + {j\;\beta\; a_{k^{\prime}}^{I}} + {\beta\; a_{k^{\prime}}^{R}} + {j\;\alpha\; a_{k^{\prime}}^{I}}} \right){\cos\left( {A - B} \right)}} +} \\{{\left( {{j\;\beta\; a_{k^{\prime}}^{R}} - {\alpha\; a_{k^{\prime}}^{I}} - {j\;\alpha\; a_{k^{\prime}}^{R}} + {\beta\; a_{k^{\prime}}^{I}}} \right){\sin\left( {A + B} \right)}} +} \\{\left( {{j\;\beta\; a_{k^{\prime}}^{R}} - {\alpha\; a_{k^{\prime}}^{I}} + {j\;\alpha\; a_{k^{\prime}}^{R}} - {\beta\; a_{k^{\prime}}^{I}}} \right){\sin\left( {A - B} \right)}}\end{bmatrix}}}}} \\{= {\frac{1}{2N}{\sum\limits_{n = 0}^{N - 1}{\sum\limits_{k^{\prime} = 0}^{N - 1}\begin{bmatrix}{{{a_{k^{\prime}}^{*}\left( {\alpha\; - \beta} \right)}{\cos\left( {A + B} \right)}} + {{a_{k^{\prime}}\left( {\alpha + \beta} \right)}{\cos\left( {A - B} \right)}} -} \\{{j\;{a_{k^{\prime}}^{*}\left( {\alpha\; - \beta} \right)}{\sin\left( {A + B} \right)}} + {j\;{a_{k^{\prime}}\left( {\alpha + \beta} \right)}{\sin\left( {A - B} \right)}}}\end{bmatrix}}}}}\end{matrix}$for k=0, 1, . . . , N−1 where * indicates complex conjugation. Reversingthe order of summation and substituting back in the values for A and B

${\hat{a}}_{k} = {{\frac{\alpha - \beta}{2N}{\sum\limits_{k^{\prime} = 0}^{N - 1}{a_{k^{\prime}}^{*}{\sum\limits_{n = 0}^{N - 1}\left\{ {{\cos\left\lbrack {2\;\pi\;{{n\left( {k^{\prime} + k} \right)}/N}} \right\rbrack} - {j\;{\sin\left\lbrack {2\;\pi\;{{n\left( {k^{\prime} + k} \right)}/N}} \right\rbrack}}} \right\}}}}} + {\frac{\alpha + \beta}{2N}{\sum\limits_{k^{\prime} = 0}^{N - 1}{a_{k^{\prime}}{\sum\limits_{n = 0}^{N - 1}\left\{ {{\cos\left\lbrack {2\;\pi\;{{n\left( {k^{\prime} - k} \right)}/N}} \right\rbrack} + {j\;{\sin\left\lbrack {2\;\pi\;{{n\left( {k^{\prime} - k} \right)}/N}} \right\rbrack}}} \right\}}}}}}$for k=0, 1, . . . , N−1. This can be simplified using the following setsof relations:

${\sum\limits_{n^{\prime} = 0}^{N - 1}{\cos\left( {2\;\pi\;{{nm}/N}} \right)}} = \left\{ \begin{matrix}N & {{m = {iN}},} & {i = {integer}} \\0 & {otherwise} & \;\end{matrix} \right.$and

${{\sum\limits_{n = 0}^{N - 1}{\sin\left( {2\mspace{14mu}{{\pi nm}/N}} \right)}} = 0},{m = {{integer}.}}$

Thus in the equation for â_(k) above, the sine terms disappear, thefirst cosine term is equal to N whenever k′=k=0 or k′+k=N and the secondcosine term is equal to N whenever k′=k:

${\hat{a}}_{k} = \left\{ \begin{matrix}{\alpha\; a_{0}} & {k = 0} \\{{\frac{\alpha + \beta}{2}a_{k}} + {\frac{\alpha - \beta}{2}a_{N - k}^{*}}} & {{k = 1},2,\ldots\mspace{14mu},{N - 1}}\end{matrix} \right.$Thus, imbalance exhibits itself as deterministic interference fromexactly one other sub-carrier. Here, this interference is labeled the IQimbalance image (or just image) and the (N−k)-th subcarrier as the imagecarrier. When the imbalance is zero, α=β=1 and â_(k)=a_(k), as expected.This signal represents the FFT output and still requires equalization tocompensate for channel distortion. The following two sections considertwo types of adaptive equalizers: the first trains over a single OFDMsymbol, possible repeated while the latter trains over multipledifferent OFDM symbols such that the effects of imbalance are averaged.

The typical mixer imbalance is 1 dB gain imbalance and 5% (or roughly5°) phase imbalance. FIG. 2 illustrates the eye diagram of a 64 QAMsystem with this typical imbalance applied only at the transmitter andno other degradations. The dark circles at the center of gray diamondsare the transmitted points and the dashed lines are the decisionboundaries. Although there are no decision errors, it is apparent thatin a noisy environment, performance is significantly degraded. FIG. 3shows a similar figure but with both the transmitter and receiversuffering 1 dB gain imbalance and 5° phase offset. The transmittedpoints again appear as dark circles. Instead of gray diamonds, there arenow gray diamonds of diamonds that overlap the decision boundaries. Thisfigure illustrates 320 symbol errors with a 30% error rate. This is anunacceptable error rate.

Equalizer Does Not Average Effects Of Imbalance

If the equalizer is trained on one OFDM symbol, or the same OFDM symbolrepeated one or more times, then the effects of imbalance will not beaveraged over time and the equalizer taps will be (assuming no noise)

${\hat{w}}_{k} = {\frac{a_{k}}{{\hat{a}}_{k}} = \frac{2}{{\alpha\left( {1 + \rho_{k}} \right)} + {\beta\left( {1 - \rho_{k}} \right)}}}$whereρ_(k) =a ^(*) _(N−k) /a _(k)is computed based on the training symbol and is thus known a priori andremains constant over the duration of the OFDM packet. The error vectordue to the imbalance image for this non-averaging equalizer is then

${\overset{\rightarrow}{N}}_{\alpha\;\beta} = {{{{\hat{a}}_{k}{\hat{w}}_{k}} - a_{k}} = {\frac{\left( {\alpha - \beta} \right)\left( {a_{N - k}^{*} - {a_{k}\rho_{k}}} \right)}{{\alpha\left( {1 + \rho_{k}} \right)} + {\beta\left( {1 - \rho_{k}} \right)}}.}}$

Using 802.11a as an example when ρ_(k)=±1, the signal to distortionratio is

${\gamma_{Imb}}_{\rho_{k} = {\pm 1}} = {{\frac{1}{2}{{\frac{\alpha + \beta}{\alpha - \beta} + \rho_{k}}}^{2}} \approx {\frac{1}{2}{\frac{\alpha + \beta}{\alpha - \beta}}^{2}}}$where the approximation is valid because the denominator approaches zeroand the numerator approaches two when the imbalance is small renderingthe ratio term much larger than ±1.

If only TX imbalance or only RX imbalance is present, then thesignal-to-distortion ratio is

${{{{\gamma_{Imb}}_{{TX}\mspace{14mu}{only}} \approx {\frac{1}{2} \times \frac{1 + {2G_{m}^{tx}\cos\;\theta_{m}^{tx}} + \left( G_{m}^{tx} \right)^{2}}{1 - {2G_{m}^{tx}\cos\;\theta_{m}^{tx}} + \left( G_{m}^{tx} \right)^{2}}}}\gamma_{Imb}}}_{{RX}\mspace{14mu}{only}} \approx {\frac{1}{2} \times \frac{1 + {2G_{m}^{rx}\cos\;\theta_{m}^{rx}} + \left( G_{m}^{rx} \right)^{2}}{1 - {2G_{m}^{rx}\cos\;\theta_{m}^{rx}} + \left( G_{m}^{rx} \right)^{2}}}$Therefore, given identical amounts of imbalance, the transmitter andreceiver introduce identical amounts of signal distortion. However, whencombined, the imbalances may ‘align’ favorably or unfavorably. Forexample, given G^(tx) _(m)=G^(rx) _(m) and θ^(tx) _(m)=θ^(rx) _(m), thecombined signal-to-distortion ratio is about 6 dB worse than at just thetransmitter or receiver. Alternatively, whenG ^(tx) _(m)=1/G ^(rx) _(m)and θ^(tx) _(m)=−θ^(rx) _(m), the combined signal-to-distortion ratio is10-15 dB better than at just the transmitter or receiver for typicalvalues of imbalance and improves considerably as the imbalancedecreases.

Equalizer Averages Effects of Imbalance

Given that typically a_(k) and a^(*) _(N−k) are uncorrelated and theaverage a^(*) _(N−k) is zero, the noise due to imbalance will also beuncorrelated with zero mean. This implies then that the taps of anadaptive frequency domain equalizer, given sufficient time and no otherimpairments, will converge toŵ_(k)=2/(α+β)Comparing these equalizer taps to those of the non-averaging equalizer,it can be observed that the effect of perfect equalizer averaging is toforce ρ_(k) to zero.

Thus, in the case of a perfectly converged, adaptive equalizer, theinterference due to the imbalance image can be expressed as an additiveerror vector:

${{\overset{\rightarrow}{N}}_{\alpha\;\beta} = {{\frac{\alpha - \beta}{\alpha + \beta}a_{N - k}^{*}\mspace{14mu}{for}\mspace{14mu} k} = 1}},2,\ldots\mspace{14mu},{N - 1}$The average signal-to-distortion ratio due to imbalance is then(assuming E[a² _(k)]=E[a² _(N−k)])

$\gamma_{Imb} = {\frac{E\left\lbrack {a_{k}}^{2} \right\rbrack}{E\left\lbrack {{\overset{\rightarrow}{N}}_{\alpha\;\beta}}^{2} \right\rbrack} = {\left( \frac{{\alpha + \beta}}{{\alpha - \beta}} \right)^{2} = \frac{\left\lbrack {1 + {G_{m}^{tx}G_{m}^{rx}{\cos\left( {\theta_{m}^{tx} - \theta_{m}^{rx}} \right)}}} \right\rbrack^{2} + \left( {{G_{m}^{tx}\sin\;\theta_{m}^{tx}} - {G_{m}^{rx}\sin\;\theta_{m}^{rx}}} \right)^{2}}{\left\lbrack {1 - {G_{m}^{tx}G_{m}^{rx}{\cos\left( {\theta_{m}^{tx} - \theta_{m}^{rx}} \right)}}} \right\rbrack^{2} + \left( {{G_{m}^{tx}\sin\;\theta_{m}^{tx}} + {G_{m}^{rx}\sin\;\theta_{m}^{rx}}} \right)^{2}}}}$

${{{{\gamma_{Imb}}_{{TX}\mspace{14mu}{only}} = \frac{1 + {2G_{m}^{tx}\cos\;\theta_{m}^{tx}} + \left( G_{m}^{tx} \right)^{2}}{1 - {2G_{m}^{tx}\cos\;\theta_{m}^{tx}} + \left( G_{m}^{tx} \right)^{2}}}\gamma_{Imb}}}_{{RX}\mspace{14mu}{only}} = \frac{1 + {2G_{m}^{rx}\cos\;\theta_{m}^{rx}} + \left( G_{m}^{rx} \right)^{2}}{1 - {2G_{m}^{rx}\cos\;\theta_{m}^{rx}} + \left( G_{m}^{rx} \right)^{2}}$This result is identical to the result when averaging is not performed,with the exception of the ½ term. Thus, the signal-to-distortion ratioin the case where the equalizer averages the effects of imbalance is 3dB better than when the equalizer trains on a single OFDM symbol andthus does not average the effects of imbalance. Given that thedistribution of a_(k) is known for each constellation type, thedistribution of the noise vectors can also be determined as a functionof the imbalance; however, this is not as important as determining theaverage noise.

The exact number of OFDM symbols over which averaging must be performedin order to average out imbalance depends on the OFDM symbolsthemselves. Two OFDM symbols are sufficient if, for example, the secondsymbol is the sign inverted equivalent of the first (i.e.a_(k)(m)=−a_(k)(m+1)).

Note that the frequency domain equalizer is not capable of compensatingfor imbalance, although allowing it to average the imbalance effects canreduce the distortion by up to 3 dB.

FIG. 4 illustrates the signal-to-distortion ratio for either gain orphase imbalance applied, individually, at either the transmitter orreceiver. The left hand vertical axis 401 represents gain imbalanceonly, in dB. The right hand vertical axis 402 represents phase imbalanceonly, in degrees. The horizontal axis 403 represents asignal-to-distortion ratio in dB. Two curves are plotted for gain 411and phase 412 imbalance. The results are the same regardless of whetherthe imbalance is at the transmitter or receiver. The distortion will beworse if both phase and gain imbalance are present or if imbalance isintroduced at both the transmitter and receiver.

Multipath Channel

The effect of multipath fading in conjunction with imbalance is nowinvestigated. It is assumed that the delay spread of the channel(including TX/RX filters) is less than the duration of the cyclic prefix(800 nsec for 802.11a) and that any symbol alignment error issufficiently small such that inter-symbol interference (ISI) isnegligible. This model does not incorporate PA non-linearities.

Recall from above that the transmitted RF signal, incorporatingtransmitter imbalance, isz(t)=y _(I)(t)cos(2πf _(c) t)−y _(Q)(t)G ^(tx) _(m)sin(2πf _(c) t+θ^(tx)_(m))The effect of the multipath channel is that the receiver sees N_(p)copies of the transmitted signal, each with its own gain G_(p), delayt_(p) and phase θ_(p), such that

${\hat{z}(t)} = {\sum\limits_{p = 0}^{N_{p} - 1}{G_{p}\left\lbrack {{{y_{I}\left( {t - t_{p}} \right)}{\cos\left( {{2\;\pi\;{f_{c}\left( {t - t_{p}} \right)}} + \theta_{p}} \right)}} - {{y_{Q}\left( {t - t_{p}} \right)}G_{m}^{tx}{\sin\left( {{2\;\pi\;{f_{c}\left( {t - t_{p}} \right)}} + \theta_{p} + \theta_{m}^{tx}} \right)}}} \right\rbrack}}$At the receiver, a carrier with a frequency offset of f₀ with respect tothe transmitter's carrier frequency mixes this signal down to baseband.Because mixing is a linear process, each of the paths can be consideredindependently as

$\begin{matrix}{{{\hat{y}}_{I,p}(t)} = \left\{ {2{G_{p}\begin{bmatrix}{{{y_{I}\left( {t - t_{p}} \right)}{\cos\left( {{2\;\pi\;{f_{c}\left( {t - t_{p}} \right)}} + \theta_{p}} \right)}} -} \\{{y_{Q}\left( {t - t_{p}} \right)}G_{m}^{tx}{\sin\left( {{2\;\pi\;{f_{c}\left( {t - t_{p}} \right)}} +} \right.}} \\\left. {\theta_{p} + \theta_{m}^{tx}} \right)\end{bmatrix}}{\cos\left( {2\;{\pi\left( {f_{c} + f_{o}} \right)}t} \right)}} \right\}_{LPF}} \\{= {G_{p}\left\lbrack {{{y_{I}\left( {t - t_{p}} \right)}{\cos\left( {{2\;\pi\; f_{o}t} + {2\;\pi\; f_{c}t_{p}} - \theta_{p}} \right)}} +} \right.}} \\\left. {{y_{Q}\left( {t - t_{p}} \right)}G_{m}^{tx}{\sin\left( {{2\;\pi\; f_{o}t} + {2\;\pi\; f_{c}t_{p}} - \theta_{p} - \theta_{m}^{tx}} \right)}} \right\rbrack\end{matrix}$

$\begin{matrix}{{{\hat{y}}_{Q,p}(t)} = \left\{ {{- 2}G_{p}{G_{m}^{rx}\begin{bmatrix}{{y_{I}\left( {t - t_{p}} \right)}{\cos\left( {2\;\pi\;{f_{c}\left( {t -} \right.}} \right.}} \\{\left. {\left. t_{p} \right) + \theta_{p}} \right) - {y_{Q}\left( {t -} \right.}} \\{\left. t_{p} \right)G_{m}^{tx}{\sin\left( {2\;\pi\;{f_{c}\left( {t -} \right.}} \right.}} \\\left. {\left. t_{p} \right) + \theta_{p} + \theta_{m}^{tx}} \right)\end{bmatrix}}{\sin\left( {{2\;{\pi\left( {f_{c} + f_{o}} \right)}t} + \theta_{m}^{rx}} \right)}} \right\}_{LPF}} \\{= {G_{p}\begin{bmatrix}{{{y_{Q}\left( {t - t_{p}} \right)}G_{m}^{tx}G_{m}^{rx}{\cos\left( {{2\;\pi\; f_{o}t} + {2\;\pi\; f_{c}t_{p}} - \theta_{p} - \theta_{m}^{tx} + \theta_{m}^{rx}} \right)}} -} \\{{y_{I}\left( {t - t_{p}} \right)}G_{m}^{rx}{\sin\left( {{2\;\pi\; f_{o}t} + {2\;\pi\; f_{c}t_{p}} - \theta_{p} + \theta_{m}^{rx}} \right)}}\end{bmatrix}}}\end{matrix}$These signals are converted to the digital domain wheren=it×f _(s) −mN _(s),f_(s) is the A/D's sample rate, m is the OFDM symbol number,N _(s)=(N+N _(cp))is the number of samples per OFDM symbol and N_(cp) is the cyclic prefixlength. Thus, the received baseband signal from the p-th path is

${{\hat{y}}_{p}\left( {n + {mN}_{s}} \right)} = {G_{p}\begin{Bmatrix}{{{y_{I}\left( {n + {mN}_{s} - {f_{s}t_{p}}} \right)}\begin{bmatrix}{{\cos\left( {{2\;{\pi\left( {n + {mN}_{s}} \right)}{f_{o}/f_{s}}} + {2\;\pi\; f_{c}t_{p}} - \theta_{p}} \right)} -} \\{j\; G_{m}^{rx}{\sin\left( {{2\;{\pi\left( {n + {mN}_{s}} \right)}{f_{o}/f_{s}}} +} \right.}} \\\left. {{2\;\pi\; f_{c}t_{p}} - \theta_{p} + \theta_{m}^{rx}} \right)\end{bmatrix}} +} \\{j\;{{y_{Q}\left( {n + {mN}_{s} - {f_{s}t_{p}}} \right)}\begin{bmatrix}{G_{m}^{tx}G_{m}^{rx}{\cos\left( {{2\;{\pi\left( {n + {mN}_{s}} \right)}{f_{o}/f_{s}}} + {2\;\pi\; f_{c}t_{p}} -} \right.}} \\{\left. {\theta_{p} - \theta_{m}^{tx} + \theta_{m}^{rx}} \right) - {j\; G_{m}^{tx}{\sin\left( {2\;{\pi\left( {n +} \right.}} \right.}}} \\\left. {{\left. {mN}_{s} \right){f_{o}/f_{s}}} + {2\;\pi\; f_{c}t_{p}} - \theta_{p} - \theta_{m}^{tx}} \right)\end{bmatrix}}}\end{Bmatrix}}$

Define four time-invariant intermediary values that are functions of theimbalance only (where α and β are the same as defined above):α=1−jG ^(rx) _(m)sin(θ^(rx) _(m)) β=G ^(tx) _(m) G ^(rx) _(m)cos(θ^(tx)_(m)−θ^(rx) _(m))+jG ^(tx) _(m)sin(θ^(tx) _(m))α₀ =−jG ^(rx) _(m)cos(θ^(rx) _(m)) β₀ =G ^(tx) _(m) G ^(rx)_(m)sin(θ^(tx) _(m)−θ^(rx) _(m))−jG ^(tx) _(m)cos (θ^(tx) _(m))Rewriting the received baseband signal in terms of these intermediaryvalues and assuming that ISI is not an issue (i.e. thatf _(s)×max(t _(p))≦N _(cp)and ignoring TX/RX filtering effects), the baseband model incorporatingimbalance, multipath and frequency offset is obtained:

${{\hat{y}}_{m}(n)} = {\sum\limits_{p = 0}^{N_{p} - 1}{G_{p}\begin{Bmatrix}{{{y_{m,I}\left( {n - {f_{s}t_{p}}} \right)}\begin{bmatrix}{{\alpha\;{\cos\left( {{2\;{\pi\left( {n + {mN}_{s}} \right)}{f_{o}/f_{s}}} + {2\;\pi\; f_{c}t_{p}} - \theta_{p}} \right)}} +} \\{\alpha_{0}{\sin\left( {{2\;{\pi\left( {n + {mN}_{s}} \right)}{f_{o}/f_{s}}} +} \right.}} \\\left. {{2\;\pi\; f_{c}t_{p}} - \theta_{p}} \right)\end{bmatrix}} +} \\{j\;{{y_{m,Q}\left( {n - {f_{s}t_{p}}} \right)}\begin{bmatrix}{\beta\;{\cos\left( {{2\;{\pi\left( {n + {mN}_{s}} \right)}{f_{o}/f_{s}}} + {2\;\pi\; f_{c}t_{p}} -} \right.}} \\{\left. \theta_{p} \right) + {\beta_{0}\;{\sin\left( {2\;{\pi\left( {n +} \right.}} \right.}}} \\\left. {{\left. {mN}_{s} \right){f_{o}/f_{s}}} + {2\;\pi\; f_{c}t_{p}} - \theta_{p}} \right)\end{bmatrix}}}\end{Bmatrix}}}$The received set of constellation points for the m-th symbols isobtained by performing the FFT on ŷ_(m,p)(n). Exploiting the linearityof the FFT, superposition can be applied (where the subscript m has beendropped for convenience)

${\hat{a}}_{k} = {{\left\{ {\sum\limits_{p = 0}^{N_{p} - 1}{{\hat{y}}_{p}(n)}} \right\}} = {\sum\limits_{p = 0}^{N_{p} - 1}{\left\{ {{\hat{y}}_{p}(n)} \right\}}}}$where

{•} is the Fourier transform. Now, rearranging ŷ_(p)(n) such that

${\hat{y}\left( {n - {f_{s}t_{p}}} \right)} = {\sum\limits_{p = 0}^{N_{p} - 1}{G_{p}\begin{Bmatrix}{\cos\left( {{2\;\pi\;{{nf}_{o}/f_{s}}} + {2\;\pi\;{mN}_{s}{f_{o}/f_{s}}} + {2\;{\pi\left( {f_{c} + f_{o}} \right)}t_{p}} -} \right.} \\{{\left. \theta_{p} \right)\left\lfloor {{\alpha\;{y_{I}(n)}} + {j\;\beta\;{y_{Q}(n)}}} \right\rfloor} + {\sin\left( {{2\;\pi\;{{nf}_{o}/f_{s}}} + {2\;\pi\;{mN}_{s}{f_{o}/f_{s}}} +} \right.}} \\{\left. {{2\;{\pi\left( {f_{c} + f_{o}} \right)}t_{p}} - \theta_{p}} \right)\left\lbrack {{\alpha_{0}{y_{I}(n)}} + {j\;\beta_{0}{y_{Q}(n)}}} \right\rbrack}\end{Bmatrix}}}$the delay property of the Fourier transform, where n₀ is non-integer,f(n−n ₀)

exp(−j2πkn ₀ /N)F(k) for k≦N/2andf(n−n ₀)

exp(−j2πn ₀(k−N)/N)F(k) for k>N/2,can be applied

${\hat{a}}_{k} = \left\{ \begin{matrix}{\sum\limits_{p = 0}^{N_{p} - 1}{{\exp\left( {{- j}\; 2\;\pi\; f_{s}t_{p}{k/N}} \right)}\left\{ {{\hat{y}}_{p}(n)} \right\}}} & {k \leq {N/2}} \\{\sum\limits_{p = 0}^{N_{p} - 1}{{\exp\left( {{- j}\; 2\;\pi\; f_{s}{{t_{p}\left( {k - N} \right)}/N}} \right)}\left\{ {{\hat{y}}_{p}(n)} \right\}}} & {k > {N/2}}\end{matrix} \right.$Recognizing that

{ŷ_(p)(n)} is the sum of two products, the Fourier transform'smodulation property can be used—if y(n)=h(n)x(n), then

${Y(k)} = {{{H(k)}*{X(k)}} = {N^{- 1}{\sum\limits_{n = 0}^{N - 1}{{H(n)}{X\left( {k - n} \right)}}}}}$where * represents circular convolution. Now defineθ=2πmN _(s) f ₀ /f _(s)+2π(f ₀ +f _(c))t _(p)−θ_(p)which is constant over each symbol, and using the definition of thediscrete Fourier transform,

${\left\{ {\cos\left( {{2\;\pi\;{{nf}_{o}/f_{s}}} + \theta} \right)} \right\}} = {\sum\limits_{n = 0}^{N - 1}{{\cos\left( {{2\;\pi\;{{nf}_{o}/f_{s}}} + \theta} \right)}{\mathbb{e}}^{{- j}\; 2\;\pi\;{{nk}/N}}}}$

${\left\{ {\sin\left( {{2\;\pi\;{{nf}_{o}/f_{s}}} + \theta} \right)} \right\}} = {\sum\limits_{n = 0}^{N - 1}{{\sin\left( {{2\;\pi\;{{nf}_{o}/f_{s}}} + \theta} \right)}{\mathbb{e}}^{{- j}\; 2\;\pi\;{{nk}/N}}}}$The Fourier transform of αy_(I)(n)+jβy_(Q)(n) was derived above and theFourier transform of α₀y_(I)(n)+jβ₀y_(Q)(n) is a trivial variation ofthis. Thus, (assuming α₀ is zero)

${\left\{ {{\hat{y}}_{p}(n)} \right\}} = {\frac{G_{p}}{2N}{\sum\limits_{n = 1}^{N - 1}\begin{Bmatrix}\left\lbrack {{\left( {\alpha + \beta} \right)a_{n}} +} \right. \\{{\left. {\left( {\alpha - \beta} \right)a_{N - n}^{*}} \right\rbrack{\sum\limits_{n^{\prime} = 0}^{N - 1}{{\cos\left( {{2\;\pi\; n^{\prime}{f_{o}/f_{s}}} + \theta} \right)}{\mathbb{e}}^{{{- j}\; 2\;\pi\;{{n^{\prime}{({k - n})}}/N}}\;}}}} +} \\\left\lbrack {{\left( {\alpha_{0} + \beta_{0}} \right)a_{n}} +} \right. \\{\left. {\left( {\alpha_{0} - \beta_{0}} \right)a_{N - n}^{*}} \right\rbrack{\sum\limits_{n^{\prime} = 0}^{N - 1}{{\sin\left( {{2\;\pi\; n^{\prime}{f_{o}/f_{s}}} + \theta} \right)}{\mathbb{e}}^{{- j}\; 2\;\pi\;{{n^{\prime}{({k - n})}}/N}}}}}\end{Bmatrix}}}$Finally, substituting back into the original equation, the final resultis obtained (fork≦N/2;see the delay property above for the modification required for thek>N/2case)

${\hat{a}}_{k} = {\frac{1}{2N}{\sum\limits_{p = 0}^{N_{p} - 1}{G_{p}{\mathbb{e}}^{{- j}\; 2\;\pi\; f_{s}t_{p}{k/N}}{\sum\limits_{n = 1}^{N - 1}\begin{Bmatrix}\left\lbrack {{\left( {\alpha + \beta} \right)a_{n}} +} \right. \\{{\left. {\left( {\alpha - \beta} \right)a_{N - n}^{*}} \right\rbrack{\sum\limits_{n^{\prime} = 0}^{N - 1}{{\cos\left( {{2\;\pi\; n^{\prime}{f_{o}/f_{s}}} + \theta} \right)}{\mathbb{e}}^{{{- j}\; 2\;\pi\;{{n^{\prime}{({k - n})}}/N}}\;}}}} +} \\\left\lbrack {{\left( {\alpha_{0} + \beta_{0}} \right)a_{n}} +} \right. \\{\left. {\left( {\alpha_{0} - \beta_{0}} \right)a_{N - n}^{*}} \right\rbrack{\sum\limits_{n^{\prime} = 0}^{N - 1}{{\sin\left( {{2\;\pi\; n^{\prime}{f_{o}/f_{s}}} + \theta} \right)}{\mathbb{e}}^{{- j}\; 2\;\pi\;{{n^{\prime}{({k - n})}}/N}}}}}\end{Bmatrix}}}}}$Again, it is assumed that the frequency offset is negligible and thuscan be assumed to be zero. In this case, the above equation simplifiesto

${\hat{a}}_{k} = \left\{ \begin{matrix}{{\frac{1}{2}{\sum\limits_{p = 0}^{N_{p} - 1}{G_{p}{\exp\left( {{- {j2\pi}}\; f_{s}t_{p}{k/N}} \right)}\begin{Bmatrix}{{{\cos(\theta)}\left\lbrack {{\left( {\alpha + \beta} \right)a_{k}} + {\left( {\alpha - \beta} \right)a_{N - k}^{*}}} \right\rbrack} +} \\{{\sin(\theta)}\left\lbrack {{\left( {\alpha_{0} + \beta_{0}} \right)a_{k}} + {\left( {\alpha_{0} - \beta_{0}} \right)a_{N - k}^{*}}} \right\rbrack}\end{Bmatrix}\mspace{14mu} 0}}} \leq k < {N/2}} \\{{\frac{1}{2}{\sum\limits_{p = 0}^{N_{p} - 1}{G_{p}{\exp\left( {{- {j2\pi}}\; f_{s}{{t_{p}\left( {k - N} \right)}/N}} \right)}\begin{Bmatrix}{{{\cos(\theta)}\left\lbrack {{\left( {\alpha + \beta} \right)a_{k}} + {\left( {\alpha - \beta} \right)a_{N - k}^{*}}} \right\rbrack} +} \\{{\sin(\theta)}\left\lbrack {{\left( {\alpha_{0} + \beta_{0}} \right)a_{k}} + {\left( {\alpha_{0} - \beta_{0}} \right)a_{N - k}^{*}}} \right\rbrack}\end{Bmatrix}\mspace{14mu}{N/2}}}} \leq k < N}\end{matrix} \right.$

To simplify the notation going forward and to assist in separating thechannel effects from the imbalance effects, the following functions aredefined (not that these are functions of the multipath channel only andare assumed constant over the duration of the packet)

$A_{\cos} = \left\{ \begin{matrix}{{\sum\limits_{p = 0}^{N_{p} - 1}{G_{p}{\exp\left( {{- {j2\pi}}\; f_{s}t_{p}{k/N}} \right)}{\cos(\theta)}\mspace{14mu} 0}} \leq k < {N/2}} \\{{\sum\limits_{p = 0}^{N_{p} - 1}{G_{p}{\exp\left( {{- {j2\pi}}\; f_{s}{{t_{p}\left( {k - N} \right)}/N}} \right)}{\cos(\theta)}\mspace{14mu}{N/2}}} \leq k < N}\end{matrix} \right.$

$A_{\sin} = \left\{ \begin{matrix}{{\sum\limits_{p = 0}^{N_{p} - 1}{G_{p}{\exp\left( {{- {j2\pi}}\; f_{s}t_{p}{k/N}} \right)}{\sin(\theta)}\mspace{14mu} 0}} \leq k < {N/2}} \\{{\sum\limits_{p = 0}^{N_{p} - 1}{G_{p}{\exp\left( {{- {j2\pi}}\; f_{s}{{t_{p}\left( {k - N} \right)}/N}} \right)}{\sin(\theta)}\mspace{14mu}{N/2}}} \leq k < N}\end{matrix} \right.$Thus, rewriting the above expression,

${\hat{a}}_{k} = {{{\frac{1}{2}\left\lbrack {{\left( {\alpha + \beta} \right)A_{\cos}} + {\left( {\alpha_{0} + \beta_{0}} \right)A_{\sin}}} \right\rbrack}a_{k}} + {{\frac{1}{2}\left\lbrack {{\left( {\alpha - \beta} \right)A_{\cos}} + {\left( {\alpha_{0} - \beta_{0}} \right)A_{\sin}}} \right\rbrack}a_{N - k}^{*}}}$

Again, assuming that a_(k) and a^(*) _(N−k) are uncorrelated, randomlydistributed and of zero mean, a frequency domain equalizer allowed toaverage over a sufficient number of OFDM symbols will converge toŵ _(k)=2/[(α+β)A _(cos)+(α₀+β₀)A _(sin)].The error vector in this case is{right arrow over (N)} _(αβ) =â _(k) ŵ _(k) −a _(k)=ε_(k) a ^(*) _(N−k)where

$ɛ_{k} = \frac{{\left( {\alpha - \beta} \right)A_{\cos}} + {\left( {\alpha_{0} - \beta_{0}} \right)A_{\sin}}}{{\left( {\alpha + \beta} \right)A_{\cos}} + {\left( {\alpha_{0} + \beta_{0}} \right)A_{\sin}}}$Notice that the error vector will be zero only if a^(*) _(N−k)=0. Theparameter, ε_(k), is of particular importance as it will be usedextensively. It is a constant value, independent of the transmitteddata, which incorporates all the channel and imbalance parameters (i.e.separates the channel and imbalance effects from the data effects).

When the frequency domain equalizer is trained on a single or repeatedOFDM reference symbol (such as for 802.11a systems), the equalizercoefficients will be (assuming no noise)

${\hat{w}}_{k} = \frac{2}{\left\lbrack {{\left( {\alpha + \beta} \right)A_{\cos}} + {\left( {\alpha_{0} + \beta_{0}} \right)A_{\sin}}} \right\rbrack + {\rho_{k}\left\lbrack {{\left( {\alpha - \beta} \right)A_{\cos}} + {\left( {\alpha_{0} - \beta_{0}} \right)A_{\sin}}} \right\rbrack}}$whereρ_(k) =a ^(*) _(N−k) /a _(k)is determined from the reference signal used for training the equalizer.In this case, the error vector is

${\overset{\rightarrow}{N}}_{\alpha\beta} = {{{{\hat{a}}_{k}{\hat{w}}_{k}} - a_{k}} = \frac{ɛ_{k}\left( {a_{N - k}^{*} - {a_{k}\rho_{k}}} \right)}{1 + {\rho_{k}ɛ_{k}}}}$

Notice that the error vector is zero whenever a_(k)ρ_(k)=a^(*) _(N−k).Using the same approach as for the no channel case, it can be shown thataveraging the imbalance effects over multiple OFDM symbols results in upto 3 dB reduction in the average distortion power.

Imbalance Compensation Algorithms

Several algorithms for compensating imbalance have been developed. Theseare focused in the following areas of pre-compensating TX imbalance atthe transmitter, compensating for TX and RX imbalance at the receiver—nochannel, compensating for TX and RX imbalance at the receiver—fadingchannel. Two variations for the fading channel case have beenconsidered: (a) compensating for TX imbalance at the receiver, ignoringRX imbalance, and (b) compensating for RX imbalance at the receiver,ignoring TX imbalance. These two variations are of interest because atthe receiver, the RX imbalance will be the same (or very slowlychanging) over all packets whereas the TX imbalance may change frompacket to packet (particularly for a base station or wireless accesspoint).

Even when imbalance compensation is applied, some residual imbalanceeffects may still exist particularly if the previously imbalancemeasurement is not accurate or the imbalance varies over time. Here,iterative imbalance compensator coefficients are computed such that thecoefficients can be updated to compensate for any such residualimbalance. Sufficiently averaging the imbalance measurements over timecould negate the need to iteratively update the coefficients. However,this would require that either the imbalance compensation not be appliedduring this measurement period or that the measurement be performedprior to any compensation. Algorithms for estimating the imbalance arepresented below.

Pre-Compensating TX Imbalance at the Transmitter

The transmit I/Q Imbalance Compensator is implemented at the transmitterand pre-compensates for I/Q imbalance introduced by the transmit mixer(see FIG. 1).

TX imbalance compensation can be applied in either the time or frequencydomain. Both are relatively simple algorithms to implement; however, ifnot all samples originate in the frequency domain, then the time-domainapproach is preferable. For example, in 802.11a, to reduce thetransmitter's wake-up latency, the short preamble is stored in memory astime-domain samples, In this case, a time-domain TX imbalancecompensator is preferred unless the preamble samples are compensatedbefore they are committed to memory.

Time-Domain Compensation

Compensating for TX I/Q imbalance in the time-domain at the transmitterrequires that y(n) be computed as a function of x(n) such thatŷ(n)=x(n), where all but the transmitter imbalance impairments are zero.Using equations derived above, this implies thatŷ _(I)(n)=y _(I)(n)−y _(Q)(n)G ^(tx) _(m)sinθ^(tx) _(m) ≡x _(I)(n)ŷ _(Q)(n)=y _(Q)(n)G ^(tx) _(m)cosθ^(tx) _(m) ≡x _(Q)(n)Solving these equations for y_(I)(n) and y_(Q)(n) as a function ofx_(I)(n) and x_(Q)(n), the functionality of the time-domain transmit I/Qimbalance compensator is determined:y _(I)(n)=x _(I)(n)+x _(Q)(n) tan (θ^(tx) _(m))y _(Q)(n)=x _(Q)(n)[G ^(tx) _(m)cos(θ^(tx) _(m))]⁻¹This assumes that G^(tx) _(m) and θ^(tx) _(m) are perfectly known andtime-invariant. However, in practice, typically only estimates of G^(tx)_(m) and θ^(tx) _(m) are known. Fortunately, one can iteratively updatethese values given new estimates. For example,G ^(tx) _(m)(i+1)=G ^(tx) _(m)(i)Ĝ ^(tx) _(m)andθ^(tx) _(m)(i+1)=θ^(tx) _(m)(i)+{circumflex over (θ)}^(tx) _(m)where Ĝ^(tx) _(m)(i) and {circumflex over (θ)}^(tx) _(m)(i) are the i-thestimates of the residual gain and phase imbalance andG ^(tx) _(m)(0)=1andθ^(tx) _(m)(0)=0.

Frequency-Domain Compensation

Because TX imbalance is introduced prior to the channel and far-endreceiver, the results of above can be used where the RX imbalance iszero. In other words

$\begin{matrix}{{{\hat{a}}_{k}}_{{TX}{\mspace{11mu}\;}{only}} = {{\frac{1 + \beta}{2}a_{k}} + {\frac{1 - \beta}{2}a_{N - k}^{*}}}} & {{k = 1},2,\ldots\mspace{14mu},{N - 1}}\end{matrix}$whereβ=G ^(tx) _(m) exp(jθ ^(tx) _(m)).If the compensator output is {tilde under (a)}_(k), then it must compute{tilde under (a)}_(k) from a_(k) such that â_(k)=a_(k). Mathematically,using the equations derived above, this implies that

${\hat{a}}_{k} = {{{\frac{1 + \beta}{2}{\overset{\sim}{a}}_{k}} + {\frac{1 - \beta}{2}{\overset{\sim}{a}}_{N - k}^{*}}} \equiv a_{k}}$

${\hat{a}}_{N - k} = {{{\frac{1 + \beta}{2}{\overset{\sim}{a}}_{N - k}} + {\frac{1 - \beta}{2}{\overset{\sim}{a}}_{k}^{*}}} \equiv a_{N - k}}$Solving these equations for {tilde under (a)}_(k) as a function ofa_(k), the functionality of the frequency domain transmit I/Q imbalancecompensator is determined:{tilde under (a)} _(k) =C ₁ a _(k) +C ₂ a ^(*) _(N−k)

$\begin{matrix}{C_{1} = {{\frac{1 + \beta^{*}}{\beta + \beta^{*}}\mspace{14mu}{and}\mspace{14mu} C_{2}} = {- \frac{1 - \beta}{\beta + \beta^{*}}}}} & \;\end{matrix}$These coefficients can be scaled by any complex value provided that bothare scaled by the same value. For example, C₁ and C₂ can be scaled by1/C₁ to simplify implementation. This will alter the eventual signalpower but given that C₁ is typically close to 1, this may be acceptable.The frequency domain equalizer at the receiver will automaticallycompensate for any such scaling.

Above, it is assumed that β is time-invariant and known exactly;however, typically only an estimate, {circumflex over (β)} is known.Thus, the compensator will not fully correct for all imbalance and someresidual imbalance effects will remain. Using this estimate in thecoefficients and computing the received points,

${{\hat{a}}_{k}}_{{TX}{\mspace{11mu}\;}{only}} = {{{\frac{1 + \beta_{\Delta}}{2}a_{k}} + {\frac{1 - \beta_{\Delta}}{2}a_{N - k}^{*}\mspace{14mu}{where}\mspace{14mu}\beta_{\Delta}}} = \frac{{\hat{\beta}}^{*} + {2\beta} - \hat{\beta}}{{\hat{\beta}}^{*} + \hat{\beta}}}$Thus, β₆₆ is the residual offset that can be compensated for by eitherconcatenating a second compensator to the original or by updating theoriginal coefficients to include the residual offset. The latterapproach is preferred as it allows a single compensator to becontinuously updated whenever a new residual TX imbalance estimate isobtained. Applying the output of the first compensator to the input ofthe second compensator and then rearranging the resulting equation toget a single compensator, the coefficient update using algorithm isobtained:

${C_{1}\left( {i + 1} \right)} = {{{C_{1}(i)}\frac{1 + {{\hat{\beta}}^{*}(i)}}{{\hat{\beta}(i)} + {{\hat{\beta}}^{*}(i)}}} - {{C_{2}^{*}(i)}\frac{1 - {\hat{\beta}(i)}}{{\hat{\beta}(i)} + {{\hat{\beta}}^{*}(i)}}}}$

${C_{2}\left( {i + 1} \right)} = {{{C_{2}(i)}\frac{1 + {{\hat{\beta}}^{*}(i)}}{{\hat{\beta}(i)} + {{\hat{\beta}}^{*}(i)}}} - {{C_{1}^{*}(i)}\frac{1 - {\hat{\beta}(i)}}{{\hat{\beta}(i)} + {{\hat{\beta}}^{*}(i)}}}}$whereC ₁(0)=1, C ₂(0)=0and {circumflex over (β)}(i) is the i-th estimate of the residual TXimbalance. This was computed by considering two concatenatedcompensators and determining the equivalent coefficients for a combinedcompensator.

Compensating For TX and RX Imbalance At The Receiver—No Channel

In this section, an algorithm is developed to compensate for imbalanceintroduced by both the far-end transmit mixer and the near-end receivemixer. It is assumed that there is no channel, enabling the use ofresults from above.

Compensation can be performed both in the time-domain andfrequency-domain. Both approaches are developed here; however, thetime-domain approach is preferred because it can be applied immediatelyafter the A/D converters, minimizing the effects of imbalance onfrequency offset estimation and symbol synchronization. The time-domainapproach may also be less complex to implement. Note, this approach isnot illustrated in FIG. 1 because in practice, there will always be somechannel, in which case it is not practical to perform TX imbalancecompensation in the time-domain.

The results of this section can be adapted for the cases of TX imbalanceor RX imbalance only. If TX imbalance is zero, then G^(tx) _(m)=1,θ^(tx) _(m)=0,α=1−jG ^(rx) _(m)sin θ^(rx) _(m)and β=G^(rx) _(m)cosθ^(rx) _(m). Similarly, if RX imbalance is zero,then G^(rx) _(m)=1, θ^(rx) _(m)=0, α=1 andβ=G ^(tx) _(m)exp(jθ^(tx) _(m))

Time-Domain Compensation

The time-domain TX/RX imbalance compensator must compute {circumflexover (x)}(n) from ŷ(n) such that {circumflex over (x)}(n)=y(n), whereall but the transmitter and receiver imbalance impairments are zero.Using the equations from above, this implies thatŷ _(I)(n)=y _(I)(n)−y _(Q)(n)G ^(tx) _(m)sin(θ^(tx) _(m))≡{circumflexover (x)}_(I)(n)−{circumflex over (x)} _(Q)(n)G ^(tx) _(m)sin(θ^(tx)_(m))ŷ _(Q)(n)=y _(Q)(n)G ^(tx) _(m) G ^(rx) _(m)cos(θ^(tx) _(m)−θ^(rx)_(m))−y _(I)(n)G ^(rx) _(m)sin(θ^(rx) _(m))≡{circumflex over (x)}_(Q)(n)G ^(tx) _(m) G ^(rx) _(m)cos(θ^(tx) _(m)−θ^(rx) _(m))−{circumflexover (x)} _(I)(n)G ^(rx) _(m)sin(θ^(rx) _(m))Solving these equations for {circumflex over (x)}_(I)(n) and {circumflexover (x)}_(Q)(n) as a function of ŷ_(I)(n) and ŷ_(Q)(n), thefunctionality of the transmit I/Q imbalance compensator is determined(i.e. by rearranging the equation for ŷ_(I)(n) to get expressions for{circumflex over (x)}_(I)(n) and {circumflex over (x)}_(Q)(n) and thensubstituting these two expressions back into the equation for ŷ_(Q)(n)and simplifying){circumflex over (x)} _(I)(n)=C _(II) ŷ _(I)(n)+C _(IQ) ŷ _(Q)(n)and{circumflex over (x)} _(Q)(n)C _(QI) ŷ _(I)(n)+C _(QQ) ŷ _(Q)(n)where the compensator coefficients are

${C_{II} = \frac{G_{m}^{tx}G_{m}^{rx}{\cos\left( {\theta_{m}^{tx} - \theta_{m}^{rx}} \right)}}{G_{m}^{tx}G_{m}^{rx}{\cos\left( \theta_{m}^{tx} \right)}{\cos\left( \theta_{m}^{rx} \right)}}},{C_{IQ} = \frac{G_{m}^{tx}{\sin\left( \theta_{m}^{tx} \right)}}{G_{m}^{tx}G_{m}^{rx}{\cos\left( \theta_{m}^{tx} \right)}{\cos\left( \theta_{m}^{rx} \right)}}}$

$\quad\begin{matrix}{{C_{QI} = \frac{G_{m}^{rx}{\sin\left( \theta_{m}^{rx} \right)}}{G_{m}^{tx}G_{m}^{rx}{\cos\left( \theta_{m}^{tx} \right)}{\cos\left( \theta_{m}^{rx} \right)}}},} \\{C_{QQ} = \frac{1}{G_{m}^{tx}G_{m}^{rx}{\cos\left( \theta_{m}^{tx} \right)}{\cos\left( \theta_{m}^{rx} \right)}}}\end{matrix}$These coefficients can be rewritten in terms of α and β as follows:

$\quad\begin{matrix}{{C_{II} = \frac{{Re}\lbrack\beta\rbrack}{{{Re}\lbrack\beta\rbrack} + {{{Im}\lbrack\alpha\rbrack}{{Im}\lbrack\beta\rbrack}}}},} \\{C_{IQ} = \frac{{Im}\lbrack\beta\rbrack}{{{Re}\lbrack\beta\rbrack} + {{{Im}\lbrack\alpha\rbrack}{{Im}\lbrack B\rbrack}}}}\end{matrix}$

$\quad\begin{matrix}{{C_{QI} = \frac{- {{Im}\lbrack\alpha\rbrack}}{{{Re}\lbrack\beta\rbrack} + {{{Im}\lbrack\alpha\rbrack}{{Im}\lbrack\beta\rbrack}}}},} \\{C_{QQ} = \frac{1}{{{Re}\lbrack\beta\rbrack} + {{{Im}\lbrack\alpha\rbrack}{{Im}\lbrack\beta\rbrack}}}}\end{matrix}$Note that the denominator can typically be ignored because it is thesame for all terms and thus the equalizer will compensate for it.

In real life, α and β will not be known exactly, but rather estimated as{circumflex over (α)} and {circumflex over (β)}. This results inresidual offset which can be estimated and characterized by {circumflexover (α)}_(Δ) and {circumflex over (β)}_(Δ). To determine how to updatethe original compensator coefficients such that the residual offset isalso compensated for, consider the equivalent scenario where a secondI/Q imbalance compensator immediately follows the first, where the firstoperates on {circumflex over (α)} and {circumflex over (β)} and thesecond on {circumflex over (α)}_(Δ) and {circumflex over (β)}_(Δ). Bymathematically combining theses two compensators such that the output ofthe first feeds into the input of the next, an equivalent set ofcoefficients can be determined. This leads to the following recursive(or iterative) format

$\quad\begin{matrix}{{{C_{II}\left( {i + 1} \right)} = \frac{{{{Re}\left\lbrack \hat{\beta(i)} \right\rbrack}{C_{II}(i)}} + {{{Im}\left\lbrack {\hat{\beta}(i)} \right\rbrack}{C_{QI}(i)}}}{{{Re}\left\lbrack {\hat{\beta}(i)} \right\rbrack} + {{{Im}\left\lbrack {\hat{\alpha}(i)} \right\rbrack}{{Im}\left\lbrack {\hat{\beta}(i)} \right\rbrack}}}},} \\{{C_{IQ}\left( {i + 1} \right)} = \frac{{{{Re}\left\lbrack {\hat{\beta}(i)} \right\rbrack}{C_{IQ}(i)}} + {{{Im}\left\lbrack {\hat{\beta}(i)} \right\rbrack}{C_{QQ}(i)}}}{{{Re}\left\lbrack {\hat{\beta}(i)} \right\rbrack} + {{{Im}\left\lbrack {\hat{\alpha}(i)} \right\rbrack}{{Im}\left\lbrack {\hat{\beta}(i)} \right\rbrack}}}}\end{matrix}$

$\quad\begin{matrix}{{{C_{QI}\left( {i + 1} \right)} = \frac{{C_{QI}(i)} - {{{Im}\left\lbrack {\hat{\alpha}(i)} \right\rbrack}{C_{II}(i)}}}{{{Re}\left\lbrack {\hat{\beta}(i)} \right\rbrack} + {{{Im}\left\lbrack {\hat{\alpha}(i)} \right\rbrack}{{Im}\left\lbrack {\hat{\beta}(i)} \right\rbrack}}}},} \\{{C_{QQ}\left( {i + 1} \right)} = \frac{{C_{QQ}(i)} - {{{Im}\left\lbrack {\hat{\alpha}(i)} \right\rbrack}{C_{IQ}(i)}}}{{{Re}\left\lbrack {\hat{\beta}(i)} \right\rbrack} + {{{Im}\left\lbrack {\hat{\alpha}(i)} \right\rbrack}{{Im}\left\lbrack {\hat{\beta}(i)} \right\rbrack}}}}\end{matrix}$whereC _(II)(0)=1, C _(IQ)(0)=0, C _(QI)(0)=0, C _(QQ)(0)=1and {circumflex over (α)}(i) and {circumflex over (β)}(i) are the i-thestimates of the residual imbalance. Again the denominator can beignored provided that the frequency domain equalizer is adaptive or theerror is deemed negligible.

Frequency-Domain Compensation

Recall from above that there are two versions of the equalized frequencydomain output depending on whether or not the equalizer averages out theeffects of imbalance by training over a sufficient number of unique OFDMsymbols. Accordingly, each version of equalizer has a correspondingTX/RX imbalance compensator algorithm.

Equalizer Does Not Average Effects Of Imbalance

The equalizer output assuming that the effects of imbalance have notbeen averaged (i.e. equalizer trained on one symbol or one repeatingsymbol) is

${\hat{\alpha}}_{k,{eq}} = \frac{{\left( {\alpha + \beta} \right)\alpha_{k}} + {\left( {\alpha - \beta} \right)\alpha_{N - k}^{*}}}{{\alpha\left( {1 + \rho_{k}} \right)} + {\beta\left( {1 - \rho_{k}} \right)}}$whereρ_(k) =a ^(x) _(N−k) /a _(k)is computed from the training symbol.

The TX/RX IQ imbalance compensator produces the compensated output{tilde under (a)}_(k) from the two inputs â_(k,eq) and â_(N−k,eq). Thecompensation algorithm can be obtained by rearranging â_(k,eq) andâ_(N−k,eq) to get two expressions for a^(*) _(N−k) which can be equatedsuch that the solution for {tilde under (a)}_(k) as a function ofâ_(k,eq) and â_(N−k,eq) can be obtained:{tilde under (a)} _(k) =C ₁ â _(k,eq) +C ₂ â ^(*) _(N−k,eq)

$C_{1} = {\frac{{\alpha\left( {1 + \rho_{k}} \right)} + {\beta\left( {1 - \rho_{k}} \right)}}{2\left( {{\alpha\;\beta^{*}} + {\beta\;\alpha^{*}}} \right)}\left( {\alpha^{*} + \beta^{*}} \right)}$and

$C_{2} = {\frac{{\alpha^{*}\left( {1 + \rho_{N - k}^{*}} \right)} + {\beta^{*}\left( {1 - \rho_{N - k}^{*}} \right)}}{2\left( {{\alpha\;\beta^{*}} + {\beta\;\alpha^{*}}} \right)}\left( {\alpha - \beta} \right)}$Notice that if the training signal is BPSK (as in 802.11a), ρ_(k)=ρ^(*)_(N−K)=±1.

In real life, α and β will be estimated as {circumflex over (α)} and{circumflex over (β)} resulting in some residual offset aftercompensation characterized by α_(Δ) and β_(Δ). This residual offset canbe compensated for by concatenating a second imbalance compensator tothe first, where the first operates on {circumflex over (α)} and{circumflex over (β)} and the second on {circumflex over (α)}_(Δ) and{circumflex over (β)}_(Δ). By mathematically combining these twocompensators such that the output of the first feeds into the input ofthe next, an equivalent set of coefficients can be determined. This ofcourse leads to the following recursive (or iterative) format

${C_{1}\left( {i + 1} \right)} = {{{C_{1}(i)}\frac{{{\hat{\alpha}(i)}\left( {1 + \rho_{k}} \right)} + {{\hat{\beta}(i)}\left( {1 - \rho_{k}} \right)}}{2\left( {{{\hat{\alpha}(i)}{{\hat{\beta}}^{*}(i)}} + {{\hat{\beta}(i)}{{\hat{\alpha}}^{*}(i)}}} \right)}\left( {{{\hat{\alpha}}^{*}(i)} + {{\hat{\beta}}^{*}(i)}} \right)} - {{C_{2}^{*}(i)}\frac{{{{\hat{\alpha}}^{*}(i)}\left( {1 + \rho_{N - k}^{*}} \right)} + {{{\hat{\beta}}^{*}(i)}\left( {1 - \rho_{N - k}^{*}} \right)}}{2\left( {{{\hat{\alpha}(i)}{{\hat{\beta}}^{*}(i)}} + {{\hat{\beta}(i)}{{\hat{\alpha}}^{*}(i)}}} \right)}\left( {{\hat{\alpha}(i)} - {\hat{\beta}(i)}} \right)}}$

${C_{2}\left( {i + 1} \right)} = {{{C_{2}(i)}\frac{{{\hat{\alpha}(i)}\left( {1 + \rho_{k}} \right)} + {{\hat{\beta}(i)}\left( {1 - \rho_{k}} \right)}}{2\left( {{{\hat{\alpha}(i)}{{\hat{\beta}}^{*}(i)}} + {{\hat{\beta}(i)}{{\hat{\alpha}}^{*}(i)}}} \right)}\left( {{{\hat{\alpha}}^{*}(i)} + {{\hat{\beta}}^{*}(i)}} \right)} - {{C_{1}^{*}(i)}\frac{{{{\hat{\alpha}}^{*}(i)}\left( {1 + \rho_{N - k}^{*}} \right)} + {{{\hat{\beta}}^{*}(i)}\left( {1 - \rho_{N - k}^{*}} \right)}}{2\left( {{{\hat{\alpha}(i)}{{\hat{\beta}}^{*}(i)}} + {{\hat{\beta}(i)}{{\hat{\alpha}}^{*}(i)}}} \right)}\left( {{\hat{\alpha}(i)} - {\hat{\beta}(i)}} \right)}}$whereC _(I)(0)=1, C₂(0)=0and {circumflex over (α)}(i) and β(i) are the i-th estimates of theresidual imbalance.

Alternatively, given that the equalizer does not average the effects ofimbalance, the equalizer taps can be adjusted such that this averagingeffect is mimicked. For example,

$w_{k}^{({ave})} = {w_{k}\frac{\alpha + \beta}{{\alpha\left( {1 - \rho_{k}} \right)} + {\beta\left( {1 + \rho_{k}} \right)}}}$This allows for the imbalance compensator based on an equalizer thataverages the effects of imbalance to be used.

Equalizer Averages Effects Of Imbalance

An equalizer that averages the frequency domain distortion due toimbalance is effectively averaging ρ_(k) such that it tends towardszero. Thus by simply forcing ρ_(k)=0,{tilde under (a)} _(k) C ₁ â _(k,eq) +C ₂ â ^(*) _(N−k,eq)

${C_{1}\left( {i + 1} \right)} = {{{C_{1}(i)}\frac{\left( {{\hat{\alpha}(i)} + {\hat{\beta}(i)}} \right)\left( {{{\hat{\alpha}}^{*}(i)} + {{\hat{\beta}}^{*}(i)}} \right)}{2\left( {{{\hat{\alpha}(i)}{{\hat{\beta}}^{*}(i)}} + {{\hat{\beta}(i)}{{\hat{\alpha}}^{*}(i)}}} \right)}} - {{C_{2}^{*}(i)}\frac{\left( {{\hat{\alpha}(i)} - {\hat{\beta}(i)}} \right)\left( {{{\hat{\alpha}}^{*}(i)} + {{\hat{\beta}}^{*}(i)}} \right)}{2\left( {{{\hat{\alpha}(i)}{{\hat{\beta}}^{*}(i)}} + {{\hat{\beta}(i)}{{\hat{\alpha}}^{*}(i)}}} \right)}}}$

${C_{2}\left( {i + 1} \right)} = {{{C_{2}(i)}\frac{\left( {{\hat{\alpha}(i)} + {\hat{\beta}(i)}} \right)\left( {{{\hat{\alpha}}^{*}(i)} + {{\hat{\beta}}^{*}(i)}} \right)}{2\left( {{{\hat{\alpha}(i)}{{\hat{\beta}}^{*}(i)}} + {{\hat{\beta}(i)}{{\hat{\alpha}}^{*}(i)}}} \right)}} - {{C_{1}^{*}(i)}\frac{\left( {{\hat{\alpha}(i)} - {\hat{\beta}(i)}} \right)\left( {{{\hat{\alpha}}^{*}(i)} + {{\hat{\beta}}^{*}(i)}} \right)}{2\left( {{{\hat{\alpha}(i)}{{\hat{\beta}}^{*}(i)}} + {{\hat{\beta}(i)}{{\hat{\alpha}}^{*}(i)}}} \right)}}}$whereC ₁(0)=1, C ₂(0)=0and {circumflex over (α)}(i) and {circumflex over (β)}(i) are the i-thestimates of the residual imbalance.

Alternatively, if the equalizer averages the effects of imbalance, theequalizer taps could be adjusted for the imbalance effects such thatthis averaging effect is removed. For example, if the averaged equalizertaps are w^((ave)) _(k), then the corresponding non-averaged taps are

$w_{k} = {w_{k}^{({ave})}\frac{{\alpha\left( {1 - \rho_{k}} \right)} + {\beta\left( {1 + \rho_{k}} \right)}}{\alpha + \beta}}$where ρ_(k) can be arbitrarily selected. This allows for the imbalancecompensator based on an equalizer that does not average imbalance to beused.

Compensating For TX and RX Imbalance At The Receiver—Fading Channel

In this section, algorithms are developed for compensating for imbalanceintroduced at the far-end transmitter and near-end receiver when afading channel is present. Thus, the equations developed in themultipath channel discussion above are used.

Time-Domain Compensation

The fading channel imparts a frequency dependency on the effects of theTX imbalance. This frequency dependency makes it very difficult, if notimpossible, to compensate for the TX imbalance in the time-domain at thereceiver. Because the RX imbalance is applied after the fading channelin the receiver, it can be removed in the time domain, but a combinedTX/RX time-domain compensator is most likely not feasible, except inlimited circumstances utilizing predetermined test symbol(s) andsubstantially noise free communications channels, which does not includethe case of a fading channel.

In particular, the effect of the multipath channel is to multiply eachsub-carrier by some complex value representing the gain and phaserotation of the channel at that sub-carrier's frequency. Because the TXimbalance is added before the fading channel and results in a linearcombination of two frequencies, in order to remove it at the receiver,the channel's effects on those two frequencies must be removed first.This implies that in order to perform time-domain imbalancecompensation, a time-domain channel equalizer must be applied first. Notonly is this very complex, but it also defeats one of the advantages ofOFDM systems and that is the ease of channel equalization in thefrequency domain.

Frequency-Domain Compensation

Both equalizer cases are considered here. In the first case, theequalizer is trained on one OFDM symbol or one repeating OFDM symbolsuch that the effects of imbalance are not averaged. In the second case,the equalizer trains over a sufficient number of OFDM symbols such thatthe effects of imbalance are averaged out.

Recall from above that the effects of TX imbalance are affected by thechannel and thus difficult to compensate in the time-domain. Theconverse is true in the frequency domain. In the frequency-domain, theequalizer removes the effects of the channel enablingfrequency-independent TX compensation; however, the equalizer appliesthe inverse channel to the RX imbalance requiring that RX imbalancecompensation be frequency-dependent.

Note that if frequency-domain compensation is used in conjunction withan adaptive equalizer (decision directed or not), the TX imbalancecompensation must be done after equalizer compensation but beforeequalizer adaptation (i.e. the imbalance compensated values should bepassed to the equalizer's update algorithm). This ensures that thechannel effects are removed prior to compensation but also ensures thatthe equalizer does not try to train on pre-compensated samples.

Equalizer Does Not Average Effects Of Imbalance

The equalizer output is (without averaging)

${\hat{\alpha}}_{k,{eq}} = {{{\hat{a}}_{k\;}w_{k}} = \frac{a_{k} + {a_{N - k}^{*}ɛ_{k}}}{1 + {\rho_{k}ɛ_{k}}}}$where

$ɛ_{k} = \frac{{\left( {\alpha - \beta} \right)A_{\cos}} + {\left( {\alpha_{0} - \beta_{0}} \right)A_{\sin}}}{{\left( {\alpha + \beta} \right)A_{\cos}} + {\left( {\alpha_{0} + \beta_{0}} \right)A_{\sin}}}$Because the A_(cos) and A_(sum) channel terms can not be separated fromthe imbalance terms, in order to compensate for the imbalance, thechannel effects will have to be accounted for. Fortunately, ε_(k) isindependent of the transmitted data and thus, in a stationary channel,will remain constant over the duration of a packet transmission.

The TX/RX IQ imbalance compensator produces the compensated output{tilde under (a)}_(k) from the two inputs, â_(k,eq) and â_(N−k,eq). Thecompensation algorithm can be obtained by rearranging â_(k,eq) andâ_(N−k,eq) to get two expressions for a^(*) _(N−k) which can the beequated such that the solution for {tilde under (a)}_(k) as a functionof â_(k,eq) and â_(N−k,eq) can be obtained:{tilde under (a)} _(k) =C ₁ â _(k,eq) +C ₂ â ^(*) _(N−k,eq)

$C_{1} = \frac{1 + {\rho_{k}ɛ_{k}}}{1 - {ɛ_{k}ɛ_{N - k}^{*}}}$and

$C_{2} = {- \frac{ɛ_{k}\left( {1 + {\rho_{N - k}^{*}ɛ_{N - k}^{*}}} \right)}{1 - {ɛ_{k}ɛ_{N - k}^{*}}}}$

In practice, ε_(k) will not be known exactly, but rather estimated as{circumflex over (ε)}_(k) resulting in some residual imbalance aftercompensation characterized by ε_(k,Δ). This residual offset can becompensated for by concatenating a second imbalance compensator to thefirst, where the first operates on {circumflex over (ε)}_(k) and thesecond on ε_(k,Δ). By mathematically combining these two compensatorssuch that the output of the first feeds into the input of the next, asingle compensator using an equivalent set of coefficients can bedetermined. The result leads into the following recursive (or iterative)format

${C_{1}\left( {i + 1} \right)} = \frac{{{C_{1}(i)}\left( {1 + {\rho_{k}{{\hat{ɛ}}_{k}(i)}}} \right)} - {{C_{2}^{*}(i)}{ɛ_{k}(i)}\left( {1 + {\rho_{N - k}^{*}{{\hat{ɛ}}_{N - k}^{*}(i)}}} \right)}}{1 - {{{\hat{ɛ}}_{k}(i)}{{\hat{ɛ}}_{N - k}^{*}(i)}}}$

${C_{2}\left( {i + 1} \right)} = \frac{{{C_{2}(i)}\left( {1 + {\rho_{k}{{\hat{ɛ}}_{k}(i)}}} \right)} - {{C_{1}^{*}(i)}{ɛ_{k}(i)}\left( {1 + {\rho_{N - k}^{*}{{\hat{ɛ}}_{N - k}^{*}(i)}}} \right)}}{1 - {{{\hat{ɛ}}_{k}(i)}{{\hat{ɛ}}_{N - k}^{*}(i)}}}$whereC ₁(0)=1, C ₂(0)=0,and {circumflex over (ε)}_(k)(i) is the i-th estimate of the combinedchannel and residual imbalance measure for the k-th subcarrier.

Equalizer Averages Effects Of Imbalance

The results from immediately above can be reused here by settingρ_(k)=0. Therefore,{tilde under (a)} _(k) =C ₁ â _(k,eq) +C ₂ â ^(*) _(N−k,eq)

${C_{1}\left( {i + 1} \right)} = \frac{{C_{1}(i)} - {{C_{2}^{*}(i)}{ɛ_{k}(i)}}}{1 - {{{\hat{ɛ}}_{k}(i)}{{\hat{ɛ}}_{N - k}^{*}(i)}}}$and

${C_{2}\left( {i + 1} \right)} = \frac{{C_{2}(i)} - {{C_{1}^{*}(i)}{ɛ_{k}(i)}}}{1 - {{{\hat{ɛ}}_{k}(i)}{{\hat{ɛ}}_{N - k}^{*}(i)}}}$whereC ₁(0)=1, C ₂(0)=0and {circumflex over (ε)}_(k)(i) is the i-th estimate of the combinedchannel and residual imbalance measure for the k-th subcarrier.

Compensating For TX Imbalance At The Receiver—Fading Channel

In this section, an algorithm is developed for compensating forimbalance introduced by the far-end transmit mixer when a fading channelis present. It is assumed that either there is no receiver imbalance orit is perfectly compensated for.

Only frequency domain TX imbalance compensation is considered here.Again, this compensation can be performed assuming either an equalizertrained on a single or repeated OFDM symbol, or an equalizer thataverages the effects of imbalance over many OFDM symbols.

Note that if frequency-domain compensation is used in conjunction withan adaptive equalizer (decision directed or not), the TX imbalancecompensation must be done after equalizer compensation but beforeequalizer adaptation (i.e. the imbalance compensated values should bepassed to the equalizer's update algorithm).

Equalizer Does Not Average Effects Of Imbalance

This case is identical to that of the combined TX/RX imbalancecompensator where the equalizer does not average effects of imbalance,except here, the RX imbalance is assumed zero. Therefore,

${ɛ_{k}}_{{TX}\mspace{14mu}{only}} = {\frac{{\left( {\alpha - \beta} \right)A_{\cos}} + {\left( {\alpha_{0} - \beta_{0}} \right)A_{\sin}}}{{\left( {\alpha + \beta} \right)A_{\cos}} + {\left( {\alpha_{0} + \beta_{0}} \right)A_{\sin}}} = \frac{1 - \beta}{1 + \beta}}$where

β_(TX  only) = G_(m)^(tx)exp (jθ_(m)^(tx))which is not a function of frequency (as expected, because the channelequalizer removes the effects of the channel). Substituting this intothe expressions for the compensator coefficients,{tilde under (a)} _(k) =C ₁ â _(k,eq) +C ₂ â ^(*) _(N−k,eq)

${C_{1}\left( {i + 1} \right)} = {{{C_{1}(i)}\frac{\left( {1 + \rho_{k}} \right) + {{\hat{\beta}(i)}\left( {1 - \rho_{k}} \right)}}{2\left( {{\hat{\beta}(i)} + {{\hat{\beta}}^{*}(i)}} \right)}\left( {1 + {{\hat{\beta}}^{*}(i)}} \right)} - {{C_{2}^{*}(i)}\frac{\left( {1 + \rho_{N - k}^{*}} \right) + {{{\hat{\beta}}^{*}(i)}\left( {1 - \rho_{N - k}^{*}} \right)}}{2\left( {{\hat{\beta}(i)} + {{\hat{\beta}}^{*}(i)}} \right)}\left( {1 - {\hat{\beta}(i)}} \right)}}$

${C_{2}\left( {i + 1} \right)} = {{{C_{2}(i)}\frac{\left( {1 + \rho_{k}} \right) + {{\hat{\beta}(i)}\left( {1 - \rho_{k}} \right)}}{2\left( {{{\hat{\beta}}^{*}(i)} + {\hat{\beta}(i)}} \right)}\left( {1 + {{\hat{\beta}}^{*}(i)}} \right)} - {{C_{1}^{*}(i)}\frac{\left( {1 + \rho_{N - k}^{*}} \right) + {{{\hat{\beta}}^{*}(i)}\left( {1 - \rho_{N - k}^{*}} \right)}}{2\left( {{\beta^{*}(i)} + {\hat{\beta}(i)}} \right)}\left( {1 - {\hat{\beta}(i)}} \right)}}$whereC ₁(0)=1, C ₂(0)=0and {circumflex over (β)}(i) is the i-th estimate of the residualfar-end TX imbalance.

Equalizer Averages Effects Of Imbalance

Recall that an equalizer that averages the frequency domain distortiondue to imbalance is effectively averaging ρ_(k) such that it tendstowards zero. Thus the previous results from immediately above can bereused here by forcing ρ_(k)=0. Therefore,{tilde under (a)} _(k) =C ₁ â _(k,eq) +C ₂ â ^(*) _(N−k,eq)

${C_{1}\left( {i + 1} \right)} = {{{C_{1}(i)}\frac{1 + {\hat{\beta}(i)}}{2\left( {{{\hat{\beta}}^{*}(i)} + {\hat{\beta}(i)}} \right)}\left( {1 + {{\hat{\beta}}^{*}(i)}} \right)} - {{C_{2}^{*}(i)}\frac{1 + {{\hat{\beta}}^{*}(i)}}{2\left( {{\beta^{*}(i)} + {\hat{\beta}(i)}} \right)}\left( {1 - {{\hat{\beta}}^{*}(i)}} \right)}}$

${C_{2}\left( {i + 1} \right)} = {{{C_{2}(i)}\frac{1 + {\hat{\beta}(i)}}{2\left( {{{\hat{\beta}}^{*}(i)} + {\hat{\beta}(i)}} \right)}\left( {1 + {{\hat{\beta}}^{*}(i)}} \right)} - {{C_{1}^{*}(i)}\frac{1 + {{\hat{\beta}}^{*}(i)}}{2\left( {{\beta^{*}(i)} + {\hat{\beta}(i)}} \right)}\left( {1 - {\hat{\beta}(i)}} \right)}}$whereC ₁(0)=1, C ₂(0)=0and {circumflex over (β)}(i) is the i-th estimate of the residualfar-end TX imbalance.

Compensating For RX Imbalance At The Receiver—Fading Channel

In this section, an algorithm is developed for compensating forimbalance introduced by the receiver mixer when a fading channel ispresent. It is assumed that either there is no transmitter imbalance orit is compensated for at a later stage in the receiver; therefore, theequations derived for the effects of multipath fading can be used whereα=1−jG ^(rx) _(m)sinθ^(rx) _(m),α₀ −jG ^(rx) _(m)cos θ^(rx) _(m),β=jα₀ and β₀=−jα.

Both time-domain and frequency domain approaches to RX imbalance areconsidered. In the frequency domain, two cases are addressed: when theequalizer does not average the imbalance error and when the equalizerdoes average the imbalance error.

Time-Domain Compensation

Because the receiver I/Q imbalance occurs after the fading channel andthe time-domain imbalance compensation occurs before channelequalization, the effects of the channel can be ignored. Thus, theresults from above can be applied here where the TX imbalance effectsare ignored. Note that because in the transceiver chain, the RX mixer isdirectly followed by the RX imbalance compensation (as opposed to havingthe TX mixer or TX imbalance compensator between the two), there is noissue about TX imbalance being present for RX imbalance compensation.

Therefore, with TX imbalance ignored,{circumflex over (x)} ₁(n)=C _(II) ŷ _(I)(n)+C _(IQ) ŷ _(Q)(n)and{circumflex over (x)} _(Q)(n)=C _(QI) ŷ _(I)(n)+C _(QQ) ŷ _(Q)(n)where the compensator coefficients are

-   -   C_(II)=1, C_(IQ)=0

${C_{QI} = {\frac{G_{m}^{rx}\sin\;\theta_{m}^{rx}}{G_{m}^{rx}\cos\;\theta_{m}^{rx}} = {\tan\;\theta_{m}^{rx}}}},{C_{QQ} = \frac{1}{G_{m}^{rx}\cos\;\theta_{m}^{rx}}}$These coefficients can be rewritten in terms of α and α₀ as follows:

-   -   C_(II)=1, C_(IQ)=0

${C_{QI} = \frac{{Im}\lbrack\alpha\rbrack}{{Im}\left\lbrack \alpha_{0} \right\rbrack}},{C_{QQ} = \frac{- 1}{{Im}\left\lbrack \alpha_{0} \right\rbrack}}$Note that the denominator can typically be ignored because the equalizerwill account for it (this requires then that C_(II)=Im[α₀]).

From the work above, with TX imbalance ignored, the iterative form isC _(II)(i+1)=C _(II)(i)=1, C _(IQ)(i+1)=C _(IQ)(i)=0

$\quad\begin{matrix}{{{C_{QI}\left( {i + 1} \right)} = \frac{{- {C_{QI}(i)}} + {{Im}\left\lbrack {\hat{\alpha}(i)} \right\rbrack}}{{Im}\left\lbrack {{\hat{\alpha}}_{0}(i)} \right\rbrack}},} \\{{C_{QQ}\left( {i + 1} \right)} = \frac{- {C_{QQ}(i)}}{{Im}\left\lbrack {{\hat{\alpha}}_{0}(i)} \right\rbrack}}\end{matrix}$whereC _(II)(0)=1, C _(IQ)(0)=0, C _(QI)(0)=0, C _(QQ)(0)=1and {circumflex over (α)}(i) and {circumflex over (α)}₀(i) are the i-thestimates of the residual imbalance. Again the denominator can beignored provided that the frequency domain equalizer is adaptive or theerror is deemed negligible (and thatC _(II)(i+1)=C _(II)(i)Im[{circumflex over (α)}₀(i)])

Frequency-Domain Compensation

The following sections derive the compensation algorithms for the twocases where the frequency domain equalizer either averages or does notaverage the imbalance error. Recall that even though the channel doesnot affect the RX imbalance, the equalizer will essentially pass the RXimbalance through the inverse of the channel making the effects of RXimbalance frequency dependent in the frequency domain.

Equalizer Does Not Average Effects Of Imbalance

This case is identical to that of the combined TX/RX imbalancecompensator where the equalizer does not average effects of imbalance,except here, the TX imbalance is assumed zero. Therefore,

${ɛ_{k}❘_{{RX}\mspace{14mu}{only}}} = {\frac{{\left( {\alpha - \beta} \right)A_{\cos}} + {\left( {\alpha_{0} - \beta_{0}} \right)A_{\sin}}}{{\left( {\alpha + \beta} \right)A_{\cos}} + {\left( {\alpha_{0} + \beta_{0}} \right)A_{\sin}}} = \frac{\left( {A_{\cos} + {jA}_{\sin}} \right)\left( {1 - {G_{m}^{rx}{\exp\left( {{+ j}\;\theta_{m}^{rx}} \right)}}} \right)}{\left( {A_{\cos} - {jA}_{\sin}} \right)\left( {1 + {G_{m}^{rx}{\exp\left( {- {j\theta}_{m}^{rx}} \right)}}} \right)}}$Because the A_(cos) and B_(sin) terms do not cancel, the imbalanceeffects cannot be separated from the channel effects and, in particular,because A_(cos) and B_(sin) vary with frequency, the effects ofimbalance appear frequency dependent. Notice that ε_(k) is independentof the transmitted data and thus, will be constant over the duration ofa packet.

Copying results from above, the imbalance compensator is{tilde under (a)} _(k) =C ₁ â _(k,eq) +C ₂ â ^(*) _(N−k,eq)

${C_{1}\left( {i + 1} \right)} = \frac{{{C_{1}(i)}\left( {1 + {\rho_{k}{{\hat{ɛ}}_{k}(i)}}} \right)} - {{C_{2}^{*}(i)}{ɛ_{k}(i)}\left( {1 + {\rho_{N - k}^{*}{{\hat{ɛ}}_{N - k}^{*}(i)}}} \right)}}{1 - {{{\hat{ɛ}}_{k}(i)}{{\hat{ɛ}}_{N - k}^{*}(i)}}}$

${C_{2}\left( {i + 1} \right)} = \frac{{{C_{2}(i)}\left( {1 + {\rho_{k}{{\hat{ɛ}}_{k}(i)}}} \right)} - {{C_{1}^{*}(i)}{ɛ_{k}(i)}\left( {1 + {\rho_{N - k}^{*}{{\hat{ɛ}}_{N - k}^{*}(i)}}} \right)}}{1 - {{{\hat{ɛ}}_{k}(i)}{{\hat{ɛ}}_{N - k}^{*}(i)}}}$whereC ₁(0)=1, C ₂(0)=0and {circumflex over (ρ)}_(k)(i) is the i-th estimate of the combinedchannel and residual RX imbalance measure for the k-th subcarrier.

Equalizer Averages Effects Of Imbalance

The results from the section immediately above can be reused here bysetting ρ_(k)=0. Therefore,{tilde under (a)} _(k) =C ₁ â _(k,eq) +C ₂ â ^(*) _(N−k,eq)

${C_{1}\left( {i + 1} \right)} = \frac{{C_{1}(i)} - {{C_{2}^{*}(i)}{ɛ_{k}(i)}}}{1 - {{{\hat{ɛ}}_{k}(i)}{{\hat{ɛ}}_{N - k}^{*}(i)}}}$and

${C_{2}\left( {i + 1} \right)} = \frac{{C_{2}(i)} - {{C_{1}^{*}(i)}{ɛ_{k}(i)}}}{1 - {{ɛ_{k}(i)}{ɛ_{N - k}^{*}(i)}}}$whereC ₁(0)=1, C ₂(0)=0and {circumflex over (ε)}_(k)(i) is the i-th estimate of the combinedchannel and residual RX imbalance measure.

Compensating For TX and RX Imbalance At The Receiver—Flat Channel

The flat channel is a special case of the fading channel where thefrequency response is essentially flat (i.e. the attenuation is constantover all subcarriers and the phase is linear). This case is ofparticular interest because during production, the channel is either aiore or a very short line of sight connection. In either case, thechannel can be modeled as flat. The present invention can be extended tothe flat channel case with a reasonable amount of further effort.

Imbalance Compensator Training

Several IQ imbalance compensation algorithms have been developed,without addressing how to obtain the parameters for those algorithms.The parameters identified included G^(tx) _(m), θ^(tx) _(m), α, β, α₀,β₀ and ε_(k).

In all cases, except for TX imbalance compensation at the receiver, theimbalance can be estimated in the time-domain by either comparing areceived training signal with its known value or by measuring andcomparing the signal statistics between the in-phase and quadraturepaths. Both these methods, but particularly the latter, requiresignificant time averaging in order to remove the effects of noise.Typically, this is fine for measuring RX imbalance because it isconstant or near constant over time; but it is more tedious for TXimbalance because it can change from packet to packet.

Estimating TX Imbalance

Recall that the received, equalized constellation point with no receiveimbalance in a fading channel is

${\hat{\alpha}}_{k,{eq}}^{({TX})} = {\frac{\alpha_{k} + {\alpha_{N - k}^{*}ɛ_{k}^{({TX})}}}{1 + {\rho_{k}ɛ_{k}^{({TX})}}} = \frac{{\alpha_{k}\left( {1 + \beta} \right)} + {\alpha_{N - k}^{*}\left( {1 - \beta} \right)}}{\left( {1 + \rho_{k}} \right) + {\beta\left( {1 - \rho_{k}} \right)}}}$for

$ɛ_{k}^{({TX})} = \frac{1 - \beta}{1 + \beta}$Rearranging,

$\beta = {- \frac{\alpha_{k} + \alpha_{N - k}^{*} - {{\hat{\alpha}}_{k,{eq}}^{({TX})}\left( {1 + \rho_{k}} \right)}}{\alpha_{k} - \alpha_{N - k}^{*} - {{\hat{\alpha}}_{k,{eq}}^{({TX})}\left( {1 - \rho_{k}} \right)}}}$or

${1 - \beta} = {2\frac{\alpha_{k} - {\hat{\alpha}}_{k,{eq}}^{({TX})}}{\alpha_{k} - \alpha_{N - k}^{*} - {{\hat{\alpha}}_{k,{eq}}^{({TX})}\left( {1 - \rho_{k}} \right)}}}$where β represents G^(tx) _(m)exp(jθ^(tx) _(m)). This yields a validresult except when {ρ_(k)=1, a_(k)=a^(*) _(N−k)} and {ρ_(k)=−1,a_(k)=−a^(*) _(N−k)} (in which case, the effects of imbalance cancel andthe above expressions for β, in the absence of noise, will evaluate to 0or ∞). If a_(k) is unknown, then a decision feedback algorithm isrequired. β can be averaged over k (sub-carriers and/or over time toreduce the effects of noise.

This β estimate can be used directly in determining the coefficients forboth the transmitter's frequency domain TX imbalance compensator and thereceiver's frequency domain TX imbalance compensator. To determine thecoefficients of the transmitter's time-domain TX imbalance compensator,G^(tx) _(m) and θ^(tx) _(m) are computed as

$G_{m}^{tx} = {{\hat{\beta}} = \sqrt{\left( {{Re}\left\lbrack \hat{\beta} \right\rbrack} \right)^{2} + \left( {{Im}\left\lbrack \hat{\beta} \right\rbrack} \right)^{2}}}$andθ^(tx) _(m)=<{circumflex over (β)}=tan⁻¹(Im[{circumflex over(β)}]/Re[{circumflex over (β)}])

This imbalance measurement is performed at the receiver and can bepassed to the transmitter either over the link (for example, duringdeployment) or over a dedicated channel (typically during production orcalibration).

In measuring β, it is assumed that there is no receiver imbalance.However, this is typically not the case and some receiver imbalance willbe present. (For calibrated production equipment, the receiver imbalanceis negligible or at least known.) Substituting â_(k,eq), which includesboth TX and RX imbalance, into the estimate for β which assumes only TXimbalance, the effect of RX imbalance on the TX imbalance measurement isdetermined

$\quad\begin{matrix}{\hat{\beta} = {{- \frac{\alpha_{k} + \alpha_{N - k}^{*} - {\frac{\alpha_{k} + {ɛ_{k}\alpha_{N - k}^{*}}}{1 + {ɛ_{k}\rho_{k}}}\left( {1 + \rho_{k}} \right)}}{\alpha_{k} - \alpha_{N - k} - {\frac{\alpha_{k} + {ɛ_{k}\alpha_{N - k}^{*}}}{1 + {ɛ_{k}\rho_{k}}}\left( {1 - \rho_{k}} \right)}}} = {\frac{1 - ɛ_{k}}{1 + ɛ_{k}} = \frac{{\beta\; A_{\cos}} + {\beta_{0}A_{\sin}}}{{\alpha\; A_{\cos}} + {\alpha_{0}A_{\sin}}}}}} \\{= \frac{\begin{matrix}{{\left( {{G_{m}^{tx}G_{m}^{rx}{\cos\left( {\theta_{m}^{tx} - \theta_{m}^{rx}} \right)}} + {j\; G_{m}^{tx}\sin\;\theta_{m}^{tx}}} \right)A_{\cos}} +} \\{\left( {{G_{m}^{tx}G_{m}^{rx}{\sin\left( {\theta_{m}^{tx} - \theta_{m}^{rx}} \right)}} - {j\; G_{m}^{tx}\cos\;\theta_{m}^{tx}}} \right)A_{\sin}}\end{matrix}}{{\left( {1 - {j\; G_{m}^{rx}\sin\;\theta_{m}^{rx}}} \right)A_{\cos}} + {\left( {{- j}\; G_{m}^{rx}\cos\;\theta_{m}^{rx}} \right)A_{\sin}}}} \\{= {{G_{m}^{tx}\cos\;\theta_{m}^{tx}\frac{{G_{m}^{rx}\cos\;\theta_{m}^{{rx}\;}A_{\cos}} - {{j\left( {1 - {j\; G_{m}^{rx}\sin\;\theta_{m}^{rx}}} \right)}A_{\sin}}}{{{- j}\; G_{m}^{rx}\cos\;\theta_{m}^{rx}A_{\sin}} + {\left( {1 - {j\; G_{m}^{rx}\sin\;\theta_{m}^{rx}}} \right)A_{\cos}}}} + {j\; G_{m}^{tx}\sin\;\theta_{m}^{tx}}}}\end{matrix}$From this, it can be seen that the RX imbalance terms do not cancel andthus will skew the estimate of the TX imbalance. Furthermore, becausethe RX imbalance effects are independent of the data pattern, averagingβ will not mitigate these effects. Of course, if RX imbalancecompensation is performed in the time-domain, then as it converges, thisTX imbalance measure will also converge.

Estimating RX Imbalance

Recall, the received equalized signal is (assuming no TX imbalance)

${\hat{\alpha}}_{k,{eq}}^{({RX})} = \frac{\alpha_{k} + {\alpha_{N - k}^{*}ɛ_{k}^{({RX})}}}{1 + {\rho_{k}ɛ_{k}^{({RX})}}}$where

$ɛ_{k}^{({RX})} = \frac{\left( {A_{\cos} + {j\; A_{\sin}}} \right)\left( {1 - {G_{m}^{rx}{\exp\left( {{+ j}\;\theta_{m}^{rx}} \right)}}} \right)}{\left( {A_{\cos} - {j\; A_{\sin}}} \right)\left( {1 + {G_{m}^{rx}{\exp\left( {{- j}\;\theta_{m}^{rx}} \right)}}} \right)}$Rearranging,

$ɛ_{k}^{({RX})} = \frac{{\hat{\alpha}}_{k,{eq}}^{({RX})} - \alpha_{k}}{\alpha_{N - k}^{*} - {{\hat{\alpha}}_{k,{eq}}^{({RX})}\rho_{k}}}$If a_(k) is unknown, then a decision feedback algorithm is required. Foreach sub-carrier, ε^((RX)) _(k) is constant and independent of the dataand thus can be averaged over time to reduce the effects of noise. Itcan then be applied directly in the receiver's frequency domain RXimbalance compensator, as illustrated above. In fact, if TX imbalance ispresent, ε_(k) will be measured which incorporates both TX and RXimbalance and one of the compensators above can be used.

A major source of error not yet addressed is that of the equalizer dueto AWGN noise. An equalizer cannot distinguish channel distortion fromnoise and thus must assume that the noise is negligible (equalizerstypically train over long periods of time such that the noise can beaveraged to negligible levels). Any equalizer error results in a gainand phase rotation on each sub-carrier. Unfortunately, this is the sameaverage effect that imbalance has on the sub-carriers making itimpossible to distinguish equalizer error from imbalance error, In orderto compensate for this error, consider the error vector (ignoring allother impairments)

${\overset{->}{N}}_{k} = {{{\hat{a}}_{k,{eq}}^{({RX})} - a_{k}} = \frac{ɛ_{k}^{({RX})}\left( {a_{N - k}^{*} - {a_{k}\rho_{k}}} \right)}{1 + {\rho_{k}ɛ_{k}^{({RX})}}}}$Notice that the imbalance error is zero whenever ρ_(k)a^(*)_(k)=a_(N−k). Thus, the equalizer error can be estimated on thosecarriers which have no imbalance error{circumflex over (Ω)}_(k,eq)=(â ^((RX)) _(k,eq) /a _(k))given ρ_(k)a^(*) _(k)=a_(N−k)and then used to estimate the imbalance using

$ɛ_{k}^{({RX})} = \frac{{{\hat{\Omega}}_{k,{eq}}^{- 1}{\hat{a}}_{k,{eq}}^{({RX})}} - a_{k}}{a_{N - k}^{*} - {{\hat{\Omega}}_{k,{eq}}^{- 1}{\hat{a}}_{k,{eq}}^{({RX})}\rho_{k}}}$given ρ_(k)a^(*) _(k)≠a_(N−k)The equalizer error is independent of the data transmitted and can beaveraged over time for each of the sub-carriers.

Unfortunately, the measured ε^((RX)) _(k) does not allow for the directdetermination of G^(rx) _(m)exp(jθ^(rx) _(m)) orα=1−jG ^(rx) _(m)sinθ^(rx) _(m)andα₀ =−jG ^(rx) _(m)cosθ^(rx) _(m).These RX imbalance measures are required for the receiver's time-domainRX imbalance compensator. In this case, ε^((RX)) _(k) can be used as ametric to train the adaptive time-domain RX imbalance compensator. Thelarger ε^((RX)) _(k), the larger the imbalance. Unfortunately, becausethe channel effects cannot be separated from the imbalance effects,averaging over frequency will be imperfect (i.e. if the ε^((RX)) _(k)are averaged as is, the various phase rotations will cause somecancellation; if the magnitudes of ε^((RX)) _(k) are averaged, thennoise will have a larger impact).

Alternatively, by multiplying εhu (RX)_(k) by its conjugate, thefollowing metric is obtained

${ɛ_{k}ɛ_{N - k}^{*}} = \frac{\left( {\alpha - \beta} \right)\left( {\alpha^{*} - \beta^{*}} \right)}{\left( {\alpha + \beta} \right)\left( {\alpha^{*} + \beta^{*}} \right)}$

${{ɛ_{k}ɛ_{N - k}^{*}}}_{{RX}\mspace{14mu}{only}} = \frac{1 - {2G_{m}^{rx}\cos\;\theta_{m}^{rx}} + {G_{m}^{rx}G_{m}^{rx}}}{1 + {2G_{m}^{rx}\cos\;\theta_{m}^{rx}} + {G_{m}^{rx}G_{m}^{rx}}}$and

${{ɛ_{k}ɛ_{N - k}^{*}}}_{{TX}\mspace{14mu}{only}} = \frac{1 - {2G_{m}^{tx}\cos\;\theta_{m}^{tx}} + {G_{m}^{tx}G_{m}^{tx}}}{1 + {2G_{m}^{tx}\cos\;\theta_{m}^{tx}} + {G_{m}^{tx}G_{m}^{tx}}}$This removes the effect of the channel in the frequency domain.Unfortunately, it weights TX and RX imbalance equally and is signinvariant to the imbalance (i.e. G^(tx) _(m) and 1/G^(tx) _(m) result inthe same value and ±θ^(tx) _(m) result in the same value). It ishowever, proportional to the imbalance and goes to zero when theimbalance goes to zero.

An alternative metric, which may be easier to compute is obtained bymultiplying the error vector {right arrow over (N)}_(k) by the conjugateof the (a^(*) _(N−k)−a_(k)ρ_(k)) term, to remove the effect of thedata's phase. This metric may be used to generate a table. The metricwill have some constant angle with a magnitude that varies with themagnitude of the data.metric=(â ^((RX)) _(k,eq) −a _(k))(a _(N−k) −a ^(*) _(k)ρ^(*) _(k))

Again, any equalizer error, Ω_(eq) can be removed by using thosesub-carriers for which the imbalance error is zero (i.e. when ρ_(k)a^(*)_(k)=a_(N=k)).{circumflex over (Ω)}_(k,eq)=(â ^((RX)) _(k,eq) /a _(k))given ρ_(k)a^(*) _(k)=a_(N−k)metric=(â ^((RX)) _(k,eq)Ω⁻¹ _(k,eq) a _(k))(a _(N−k) −a ^(*) _(k)ρ^(*)_(k))given ρ_(k)a^(*) _(k)≠a_(N−k)

Note that the metric is still a function of the sub-carrier index, k andfor each sub-carrier, the metric will have a unique phase. Thus,averaging these sub-carriers will be sub-optimal.

Application to IEEE 802.11a

IEEE 802.11a specifies an OFDM-based transceiver for wireless LANapplications. In order to meet the performance objectives of 802.11a,either the TX and TX imbalance must be designed to be very small orimbalance compensation must be applied.

In this section, the balance requirements are defined, imbalancecompensation is proposed and simulation results are provided.

Balance Requirements

The appendix presents calculations of imbalance criteria that flow fromthe 802.11a standard.

Imbalance Compensation Proposal

Several imbalance compensation algorithms were presented above. Althoughall of these are applicable to 802.11a, a subset can be selected baseprimarily on system configuration and with the goal of minimizingimplementation complexity.

Transmitter TX Imbalance Compensation

It is proposed that the transmitter precompensate for TX imbalance inthe time-domain using the compensator derived for compensating for TXimbalance at the transmitter in the time-domain. This should be one ofthe last blocks after filtering and PA predistortion (if applicable) andbefore the D/A's. This allows the storage of the uncompensated shorttraining symbols in the time-domain and eliminates any quantization orsaturation issues, which might arise if the imbalance were compensatedfor in the frequency domain.

It is proposed that the transmitter's TX imbalance compensator betrained during production, prior to predistorter training and transmitlevel calibrations using the algorithm derived for estimating TXimbalance. In this case, the production receiver would make themeasurement of G^(tx) _(m) and θ^(tx) _(m) and relay these back to thetransmitter under test. To minimize effort, the measurement would beperformed on a BPSK packet using the typical 802.11a packet conventions.The production receiver would synchronize and train its frequency-domainequalizer on the long symbols and then measure the TX imbalance onsubsequent OFDM symbols, averaging over frequency and time. It isnecessary that the production receiver have little or no RX imbalance(either by component design, calibration or by RX imbalancecompensation).

It is also possible to train the TX imbalance compensator duringdeployment (real-time). This would require some additional MACfunctionality such that the G^(tx) _(m) and θ^(tx) _(m) measurementscould be relayed back to the transmitter. This would be particularlyuseful if it is found that TX imbalance wanders with time.

Although a feedback circuit could be used where the TX mixer output isfeed back into the RX mixer and RX A/D's and then processed in thedigital baseband, this is considered too complex.

Receiver TX Imbalance Compensation

It is proposed that the receiver compensate for TX imbalance in thefrequency-domain using the compensator for an equalizer that does notaverage effects of imbalance. This compensator should follow theequalizer compensation but should precede the equalizer training (ifadaptive). Compensation should be disabled during the L1 and L2 trainingsymbols to enable the equalizer to attempt to compensate for theimbalance (recall, it was assumed that the equalizer did not average theimbalance over many symbols).

It is proposed that the TX imbalance be measured based on the signalfield and averaged over all sub-carriers such that the effect of noiseis reduced (i.e. averaging over 48 sub-carriers will reduce the noise byabout 17 dB) using the algorithm for estimating TX imbalance. The signalfield is unknown at the receiver and thus a hard decision is required.It is speculated that because the signal field is BPSK, in lower noiseenvironments, the probability of a hard decoding error is negligible andthus the TX imbalance estimate will be accurate. In noisy environments,the probability of a hard decoding error will increase resulting in aless accurate TX imbalance estimate. This is acceptable in that in anoisy environment, it is not the distortion due to imbalance thatdominates the noise. Regardless, the performance of the estimate can beimproved by applying a weighted average where the weighting factor isthe channel state information (CSI), computed as the reciprocal of themagnitude of the equalizer taps.

The TX imbalance measurement will typically be performed for each newpacket. Of course, if it is known which transmitter will be transmittingnext, TX imbalance compensation can be applied on the signal field and aresidual TX imbalance measurement made. Based on this, the compensatorcoefficients can be updated and applied immediately thereafter.

Receiver RX Imbalance Compensation

It is proposed that for 802.11a, receiver RX imbalance compensation beperformed in the time-domain as soon after the A/D's as possible usingthe time-domain compensator above. Performing RX compensation in thetime-domain ensures that the RX imbalance effects are removed prior tothe synchronization and equalization routines as well as before any TXimbalance measurements. Furthermore, the time-domain compensator is lesscomplex to implement than its frequency-domain counterpart. RX imbalancecompensation is applied 100% of the time.

It is proposed that the compensator be trained using an LMS type offeedback algorithm according to the discussion of receive imbalancecompensation in the frequency domain.

If multiple receive radios are used, a separate set of coefficients canbe stored for each radio. For selection diversity type of systems, onlyone radio is used at a time and the coefficients associated with thatradio can be applied. If a more advanced diversity system is used, itmay be necessary to combine the coefficients.

Performance Simulations

The imbalance compensation algorithms above are capable of compensatingfor all TX and RX imbalance. Given a perfect estimate, these algorithmsshould be able to perfectly compensate. In implementation, quantizationand perhaps some approximations will be applied in which case theperformance of the compensation will no longer be perfect; however, thisis an implementation issue and is not addressed here. The performanceaspect of imbalance compensation is in determining the coefficients forcompensation. Even here, it is possible to get very near optimalperformance depending on the system configuration. For example, for mostsystems, if not all, the RX imbalance will be constant from packet topacket enabling considerable averaging over time to remove noise andobtain very accurate compensator coefficients. If some form of pointcoordination function is used, then the same applies to the TX imbalancecompensation at the receiver. A point coordination function coordinateswhich transmitter will be active at a particular time. A base station,for instance, may keep track of imbalance compensation coefficients forseveral remote transmitters and look up which coefficients to use,corresponding to the transmitter that will be active. Furthermore, TXimbalance estimates can be averaged over multiple packets for one ormore remote transmitters, using point coordination. Nonetheless, in thissection, tolerable levels of imbalance for various typical OFDMsub-carrier constellations are determined and also the performance ofsome of the coefficient training algorithms is measured.

Imbalance Measures

In a preceding section regarding estimating RX imbalance, two metrics ofimbalance were discussed. For the alternative metric, which may beeasier to compute, the following data was compiled by simulation. Inthis table, rows are organized by the signal-to-noise ratio measured atthe receiver. The columns indicate extents of imbalance, in dB gainimbalance and degrees phase imbalance. In the following table, resultsof a simulation are tabulated. The simulation was run for 400 OFDMsymbols, Quadrature Phase Shift Keying (QPSK) and random fadingchannels. Errors were measured and averaged over 10 simulation runs. Thetable reports the minimum→maximum range, the mean and a measure ofvariance from the simulation runs.

1 dB, 0.5 dB, 0.25 dB No 5 degrees 2.5 degrees 1.25 degrees imbalance 10dB SNR 0.1583->0.2080 0.1061->0.1399 0.0659->0.1048 0.0712->0.0930 m =0.18827 m = 0.12156 m = 0.08951 m = 0.08250 s = 0.01557391 s =0.01103965 s = 0.01057917 s = 0.00873155 20 dB SNR 0.1927->0.20760.1001->0.1113 0.0522->0.0628 0.0206->0.0327 m = 0.20015 m = 0.10421 m =0.05506 m = 0.02573 s = 0.00517778 s = 0.00357754 s = 0.00359017 s =0.00389445 30 dB SNR 0.1956->0.2035 0.0980->0.1032 0.0494->0.05290.0056->0.0088 m = 0.19932 m = 0.10073 m = 0.05092 m = 0.00777 s =0.00232178 s = 0.00147652 s = 0.00112428 s = 0.00086801

Rows in the table are organized by SNR at the receiver:receive signaldivided by the noise level at the receiver. Columns are organized byvarious gain and phase imbalance combinations. From this table, we seethat the metric grows as the imbalance gets worse. We can estimate thismetric, even when it is extremely noisy. This metric converges and has asmall variance.

While the present invention is disclosed by reference to the preferredembodiments and examples detailed above, it is understood that theseexamples are intended in an illustrative rather than in a limitingsense. Computer-assisted processing is implicated in the describedembodiments. Accordingly, the present invention may be embodied inmethods for computer-assisted processing, systems including logic toimplement the methods, media impressed with logic to carry out themethods, data streams impressed with logic to carry out the methods, orcomputer-accessible processing services. It is contemplated thatmodifications and combinations will readily occur to those skilled inthe art, which modifications and combinations will be within the spiritof the invention and the scope of the following claims.

APPENDIX: IMBALANCE REQUIREMENTS FOR 802.11A

It is desirable to limit the effects of phase (±δφ) and amplitude (±ε)mismatches in I and Q to 5 dB or less below the quantization floor ofthe A/D converter used in a system. This assists in rejection of errorsthat otherwise might be introduced by phase and amplitude imbalance. Thederivations from the case of no channel and a perfectly trained adaptiveequalizer can be used to consider the issue. For example, assumingimbalance at the transmit side only and given ε=1.41 and 0^(tx) _(m)=0,the signal-to distortion ratio can be computed asG ^(tx) _(m)=20log₁₀(1+2ε/100)=0.2416 dB

${\gamma_{lmb}}_{{TX}\mspace{14mu}{only}} = {\frac{1 + {2G_{m}^{tx}\cos\;\theta_{m}^{tx}} + \left( G_{m}^{tx} \right)^{2}}{1 - {2G_{m}^{tx}\cos\;\theta_{m}^{tx}} + \left( G_{m}^{tx} \right)^{2}} = {37.14\mspace{14mu}{dB}}}$Similarly for the phase, given G^(tx) _(m)=0 and δφ=0.82 degrees

-   -   θ^(tx) _(m)=2δφ=1.64 degrees

${\gamma_{lmb}}_{{TX}\mspace{14mu}{only}} = {\frac{1 + {2G_{m}^{tx}\cos\;\theta_{m}^{tx}} + \left( G_{m}^{tx} \right)^{2}}{1 - {2G_{m}^{tx}\cos\;\theta_{m}^{tx}} + \left( G_{m}^{tx} \right)^{2}} = {36.89\mspace{14mu}{{dB}.}}}$

When the gain and phase imbalance are considered together, the signalto-distortion ratio drops to 34 dB or about 3 dB worse. When both thetransmitter and receiver have this imbalance (i.e. worst-case, aligned),the signal-to-distortion ratio drops to 28 dB. For 802.11a, theequalizer is typically trained on a single OFDM symbol repeated twiceand thus the equalizer will not train perfectly and an additional 3 dBpenalty is expected. These calculations provide guidance as to theimpact of imbalance.

1. A method of compensating for transmitter imbalance in a multi-carrier, ODFM symbol transmission system, including: estimating transmitter gain and phase imbalance by measuring signals on a plurality of carriers of the multi-carrier system, wherein estimating includes deducing constellation points transmitted from a received signal; and calculating one or more differences between the received signal and an intended signal corresponding to the deduced constellation points; and determining compensation parameters from the imbalance; loading the compensation parameters into the transmitter, whereby the compensation parameters can be used to compensate for transmitter gain and phase imbalance.
 2. The method of claim 1, wherein the estimating step is a frequency domain estimation.
 3. The method of claim 1, wherein the estimating step is a frequency domain estimation performed after processing a received signal through an equalizer.
 4. The method of claim 3, wherein a plurality but fewer than all of the carriers of the multicarrier system are utilized in the estimating step.
 5. The method of claim 1, wherein the estimating step is a frequency domain estimation such that: ${\hat{a}}_{k} = \left\{ {\begin{matrix} {{\frac{1}{2}{\sum\limits_{p = 0}^{N_{p} - 1}{G_{p}{\exp\left( {{- j}\; 2\;\pi\; f_{s}t_{p}{k/N}} \right)}\begin{Bmatrix} {{{\cos(\theta)}\left\lbrack {{\left( {\alpha + \beta} \right)a_{k}} + {\left( {\alpha - \beta} \right)a_{N - k}^{*}}} \right\rbrack} +} \\ {{\sin(\theta)}\left\lbrack {{\left( {\alpha_{0} + \beta_{0}} \right)a_{k}} + {\left( {\alpha_{0} - \beta_{0}} \right)a_{N - k}^{*}}} \right\rbrack} \end{Bmatrix}\mspace{14mu} 0}}} \leq k < {N/2}} \\ {{\frac{1}{2}{\sum\limits_{p = 0}^{N_{p} - 1}{G_{p}{\exp\left( {{- j}\; 2\;\pi\; f_{s}{{t_{p}\left( {k - N} \right)}/N}} \right)}\begin{Bmatrix} {{{\cos(\theta)}\left\lbrack {{\left( {\alpha + \beta} \right)a_{k}} + {\left( {\alpha - \beta} \right)a_{N - k}^{*}}} \right\rbrack} +} \\ {{\sin(\theta)}\left\lbrack {{\left( {\alpha_{0} + \beta_{0}} \right)a_{k}} + {\left( {\alpha_{0} - \beta_{0}} \right)a_{N - k}^{*}}} \right\rbrack} \end{Bmatrix}\mspace{14mu}{N/2}}}} \leq k < N} \end{matrix},\mspace{14mu}\text{where}} \right.$ â_(k) is a received value of a constellation point, k being the carrier index among the plurality of carriers; a_(k) is a transmitted value of the constellation point for a kth carrier; a^(*) _(N−k) is a complex conjugate of the transmitted value of the constellation point for an N−k th carrier, where N is a total number of carriers; f_(s) is a sampling rate of an analog to digital converter in a receiver used to process the received value; t_(p) is a delay measure for the pth carrier, p being an additional carrier index among the plurality of carriers; α, α₀, β and β₀ are intermediary values treated as time-invariant: α=1−jG ^(rx) _(m)sin(θ^(rx) _(m)) β=G^(tx) _(m) G ^(rx) _(m)cos(θ^(tx) _(m)−θ^(rx) _(m))+jG ^(tx) _(m)sin (θ^(tx) _(m)) α₀ =−jG ^(rx) _(m)cos(θ^(rx) _(m)) β₀ =G ^(tx) _(m) G ^(rx) _(m)sin(θ^(tx) _(m)−θ^(rx) _(m))−jG ^(tx) _(m)cos(θ^(tx) _(m)) G^(rx) _(m) is gain imbalance of a receiver's mixer; G^(tx) _(m) is gain imbalance of the transmitter's mixer; θ^(rx) _(m) is phase imbalance of the receiver's mixer; and θ^(tx) _(m) is gain imbalance of the transmitter's mixer.
 6. The method of claim 1, wherein the estimating step is a frequency domain estimation performed after processing a received signal through an equalizer, such that: $G_{m}^{tx} = {{\hat{\beta}} = \sqrt{\left( {{Re}\left\lbrack \hat{\beta} \right\rbrack} \right)^{2} + \left( {{Im}\left\lbrack \hat{\beta} \right\rbrack} \right)^{2}}}$ and θ^(tx) _(m)=<{circumflex over (β)}=tan⁻¹(Im[{circumflex over (β)}]/Re[{circumflex over (β)}]), where $\hat{\beta} = {- \;\frac{a_{k} + a_{N - k}^{*} - {{\hat{a}}_{k,{eq}}^{({TX})}\left( {1 + \rho_{k}} \right)}}{a_{k} - a_{N - k}^{*} - {{\hat{a}}_{k,{eq}}^{({TX})}\left( {1 - \rho_{k}} \right)}}}$ and G^(tx) _(m) is gain imbalance of the transmitter's mixer; θ^(tx) _(m) is gain imbalance of the transmitter's mixer; {circumflex over (β)} is an estimated value; â^((TX)) _(k,eq) is a received value of a constellation point, including TX transmitter imbalance, after the processing the received signal through the equalizer, k being the carrier index among the multiple carrier channels; a_(k) is a transmitted value of the constellation point for a k th carrier; a^(*) _(N−k) is the complex conjugate of the transmitted value of the constellation point for an N−k th carrier, where N is a total number of carriers; ρ_(k) is a ratio ρ_(k) =a ^(*) _(N−k) /a _(k) of transmitted symbols used for equalizer training.
 7. The method of claim 1 further including communicating from a receiver to the transmitter the receiver's estimates of the transmitter imbalance.
 8. The method of claim 1, wherein the estimating step includes measuring the imbalances prior to delivery of the system to a customer, by analyzing a received signal corresponding to one or more predetermined training packets transmitted according to standard packet conventions.
 9. The method of claim 8, wherein the predetermined training packets include a second symbol that is a sign inverted equivalent of a first symbol.
 10. The method of claim 1, wherein the estimating step further includes: transmitting one or more predetermined training packets according to standard packet conventions through a substantially noise-free communication channel; and performing a time-domain analysis of received packets without processing the received packets through an equalizer.
 11. The method of claim 10, wherein the predetermined training packets include a second symbol that is a sign inverted equivalent of a first symbol.
 12. The method of claim 1, wherein the transmitter is adapted to perform time domain compensation for the imbalances, such that: y _(I)(n)=x _(I)(n)+x _(Q)(n)tan(θ^(tx) _(m)) and y _(Q)(n)=x _(Q)(n)[G ^(tx) _(m)cos(θ^(tx) _(m))]⁻¹ where x_(I)(n) is an in-phase component of a signal being compensated; x_(Q)(n) is a quadrature component of the signal being compensated; y_(I)(n) is an in-phase component of the signal after being compensated; y_(Q)(n) is a quadrature component of the signal after being compensated; G^(tx) _(m) is an estimate of the gain imbalance; and θ^(tx) _(m) is an estimate of the phase imbalance.
 13. The method of claim 12, wherein the compensation parameters are determined iteratively, from a starting estimate, such that: G ^(tx) _(m)(i+1)=G ^(tx) _(m)(f)Ĝ ^(tx) _(m) and θ^(tx) _(m)(i+1)=θ^(tx) _(m)(i)+{circumflex over (θ)}^(tx) _(m), where Ĝ^(tx) _(m)(f) and {circumflex over (θ)}^(tx) _(m)(i) are i-th estimates of residual gain and phase imbalance.
 14. The method of claim 1, wherein the transmitter is adapted to perform frequency domain compensation for the imbalances, such that: ${{\hat{a}}_{k}}_{{TX}\mspace{14mu}{only}} = {{\frac{1 + \beta_{\Delta}}{2}a_{k}} + {\frac{1 - \beta_{\Delta}}{2}a_{N - k}^{*}}}$ where â_(k)|_(TX only) is a received value of a constellation point, including TX transmitter imbalance, after the processing the received signal through the equalizer, k being the carrier index among the multiple carrier channels; a_(k) is a transmitted value of the constellation point for a k th carrier; a^(*) _(N−k) is the complex conjugate of the transmitted value of the constellation point for an N−k th carrier, where N is a total number of carriers; and β_(Δ) is a residual offset that is compensated for by either concatenating a second compensator to the original or by updating the original coefficients to include the residual offset.
 15. The method of claim 14, wherein the compensation parameters are determined iteratively, from a starting estimate, such that: ${C_{1}\left( {i + 1} \right)} = {{{C_{1}(i)}\;\frac{1 + {{\hat{\beta}}^{*}(i)}}{{\hat{\beta}(i)} + {{\hat{\beta}}^{*}(i)}}} - {{C_{2}^{*}(i)}\;\frac{1 - {\hat{\beta}(i)}}{{\hat{\beta}(i)} + {{\hat{\beta}}^{*}(i)}}}}$ ${C_{2}\left( {i + 1} \right)} = {{{C_{2}(i)}\;\frac{1 + {{\hat{\beta}}^{*}(i)}}{{\hat{\beta}(i)} + {{\hat{\beta}}^{*}(i)}}} - {{C_{1}^{*}(i)}\;\frac{1 - {\hat{\beta}(i)}}{{\hat{\beta}(i)} + {{\hat{\beta}}^{*}(i)}}}}$ where C ₁(0)=1, C ₂(0)=0 and {circumflex over (β)}(i) is an i-th estimate of residual TX imbalance.
 16. A method of compensating for transmitter imbalances in a multi-carrier, OFDM symbol transmission system, including: estimating transmitter gain and phase imbalance by measuring signals on a plurality of carriers of the multi-carrier system, wherein estimating includes analyzing a received signal corresponding to one or more predetermined training packets transmitted according to standard packet conventions; and compensating in the transmitter for the transmitter imbalance responsive to the estimated transmitted gain and phase imbalance.
 17. A method of compensating for receiver imbalance in a multi-carrier, OFDM symbol transmission system, including: estimating receiver gain and phase imbalance by measuring signals on a plurality of carriers of the multi-carrier system, utilizing a transmitter having known imbalance characteristics, wherein estimating includes measuring the imbalances prior to delivery of the system to a customer, by analyzing a received signal corresponding to one or more predetermined training packets transmitted according to standard packet conventions; determining compensation parameters from the receiver imbalances; and loading the compensating parameters in the receiver to be used to compensate for receiver gain and phase imbalance.
 18. The method of claim 17, wherein the estimating step is a frequency domain estimation.
 19. The method of claim 17, wherein the estimating step is a frequency domain estimation performed after processing a received signal through an equalizer.
 20. The method of claim 19, wherein a plurality but fewer than all of the multiple carrier channels are utilized in the estimating step.
 21. The method of claim 17, wherein the estimating step is a frequency domain estimation, such that: ${\hat{a}}_{k} = \left\{ {\begin{matrix} {{\frac{1}{2}{\sum\limits_{p = 0}^{N_{p} - 1}{G_{p}{\exp\left( {{- j}\; 2\;\pi\; f_{s}t_{p}{k/N}} \right)}\begin{Bmatrix} {{{\cos(\theta)}\left\lbrack {{\left( {\alpha + \beta} \right)a_{k}} + {\left( {\alpha - \beta} \right)a_{N - k}^{*}}} \right\rbrack} +} \\ {{\sin(\theta)}\left\lbrack {{\left( {\alpha_{0} + \beta_{0}} \right)a_{k}} + {\left( {\alpha_{0} - \beta_{0}} \right)a_{N - k}^{*}}} \right\rbrack} \end{Bmatrix}\mspace{14mu} 0}}} \leq k < {N/2}} \\ {{\frac{1}{2}{\sum\limits_{p = 0}^{N_{p} - 1}{G_{p}{\exp\left( {{- j}\; 2\;\pi\; f_{s}{{t_{p}\left( {k - N} \right)}/N}} \right)}\begin{Bmatrix} {{{\cos(\theta)}\left\lbrack {{\left( {\alpha + \beta} \right)a_{k}} + {\left( {\alpha - \beta} \right)a_{N - k}^{*}}} \right\rbrack} +} \\ {{\sin(\theta)}\left\lbrack {{\left( {\alpha_{0} + \beta_{0}} \right)a_{k}} + {\left( {\alpha_{0} - \beta_{0}} \right)a_{N - k}^{*}}} \right\rbrack} \end{Bmatrix}\mspace{14mu}{N/2}}}} \leq k < N} \end{matrix},\mspace{14mu}\text{where}} \right.$ â_(k) is a received value of a constellation point, k being the carrier index among the plurality of carriers; a_(k) is a transmitted value of the constellation point for a kth carrier; a^(*) _(N−k) is a complex conjugate of the transmitted value of the constellation point for an N−k th carrier, where N is a total number of carriers; f_(t) is a sampling rate of an analog to digital converter in a receiver used to process the received value; t_(p) is a delay measure for the pth carrier, p being an additional carrier index among the plurality of carriers; α, α₀, β and β₀ are intermediary values treated as time-invariant: α=1−jG ^(rx) _(m)sin(θ^(rx) _(m)) β=G ^(tx) _(m) G ^(rx) _(m)cos(θ^(tx) _(m)−θ^(rx) _(m))+jG ^(tx) _(m)sin (θ^(tx) _(m)) α ₀ =−jG ^(rx) _(m)cos(θ^(rx) _(m)) β₀ =G ^(tx) _(m) G ^(rx) _(m)sin(θ^(tx) _(m)−θ^(rx) _(m))−jG ^(tx) _(m)cos(θ^(tx) _(m)) G^(rx) _(m) is gain imbalance of a receiver's mixer; G^(tx) _(m) is gain imbalance of the transmitter's mixer; θ^(rx) _(m) is phase imbalance of the receiver's mixer; and θ^(tx) _(m) is gain imbalance of the transmitter's mixer.
 22. The method of claim 17, wherein the estimating step is a frequency domain estimation performed after processing a received signal through an equalizer, such that: a metric ε_(k)ε^(*) _(N−k)|_(RX only) is substantially minimized, ${{ɛ_{k}ɛ_{N - k}^{*}}}_{{RX}\mspace{14mu}{only}} = \frac{1 - {2G_{m}^{rx}\cos\;\theta_{m}^{rx}} + {G_{m}^{rx}G_{m}^{rx}}}{1 + {2G_{m}^{rx}\cos\;\theta_{m}^{rx}} + {G_{m}^{rx}G_{m}^{rx}}}$ and ${ɛ_{k}^{({RX})} = \frac{{\hat{a}}_{k,{eq}}^{({RX})} - a_{k}}{a_{N - k}^{*} - {{\hat{a}}_{k,{eq}}^{({RX})}\rho_{k}}}},$ where â_(k) is a received value of a constellation point, k being the carrier index among the plurality of carriers; a_(k) is a transmitted value of the constellation point for a kth carrier; a^(*) _(N−k) is a complex conjugate of the transmitted value of the constellation point for an N−k th carrier, where N is a total number of carriers; G^(rx) _(m) is gain imbalance of a receiver's mixer; θ^(rx) _(m) is phase imbalance of the receiver's mixer; and ρ_(k) is a ratio ρ_(k)=a^(*) _(N−k)/a_(k) of transmitted symbols used for equalizer training.
 23. The method of claim 17, wherein the estimating step further includes: deducing constellation points transmitted from a received signal; and calculating one or more differences between the received signal and an intended signal corresponding to the deduced constellation points.
 24. The method of claim 17, wherein at the predetermined training packets includes a second symbol that is a sign inverted equivalent of a first symbol.
 25. The method of claim 17, wherein the estimating step further includes: transmitting one or more predetermined packets according to standard packet conventions through a substantially noise-free communications channel; and performing a time domain analysis of received packets without processing the received packets through an equalizer.
 26. The method of claim 25, wherein at the predetermined training packets includes a second symbol that is a sign inverted equivalent of a first symbol.
 27. The method of claim 17, wherein the receiver is adapted to compensate for the imbalances, such that {circumflex over (x)} _(I)(n)=C _(II) ŷ _(I)(n)+C _(IQ) ŷ _(Q)(n) and {circumflex over (x)} _(Q)(n)=C _(QI) ŷ _(I)(n)+C _(QQ) ŷ _(Q)(n) where the compensator coefficients are C_(II)=1, C_(IQ)=0 ${C_{QI} = {\frac{G_{m}^{rx}\sin\;\theta_{m}^{rx}}{G_{m}^{rx}\cos\;\theta_{m}^{rx}} = {\tan\;\theta_{m}^{rx}}}},{C_{QQ} = \frac{1}{G_{m}^{rx}\cos\;\theta_{m}^{rx}}},$ where x_(I)(n) is an in-phase component of a signal being compensated; x_(Q)(n) is a quadrature component of the signal being compensated; y_(I)(n) is an in-phase component of the signal after being compensated; y_(Q)(n) is a quadrature component of the signal after being compensated; â_(k) is a received value of a constellation point, k being the carrier index among the plurality of carriers; a_(k) is a transmitted value of the constellation point for a kth carrier; a^(*) _(N−k) is a complex conjugate of the transmitted value of the constellation point for an N−k th carrier, where N is a total number of carriers; G^(rx) _(m) is an estimate of gain imbalance; and θ^(rx) _(m) is an estimate of phase imbalance.
 28. The method of claim 27, wherein the compensation parameters are determined iteratively, from a starting estimate, such that: C _(II)(i+1)=C _(II)(i)=1, C _(IQ)(i+1)=C _(IQ)(f)=0 ${{C_{QI}\left( {i + 1} \right)} = \frac{{- {C_{QI}(i)}} + {{Im}\left\lbrack {\hat{\alpha}(i)} \right\rbrack}}{{Im}\left\lbrack {{\hat{\alpha}}_{ij}(i)} \right\rbrack}},{{C_{QQ}\left( {i + 1} \right)} = \frac{- {C_{QQ}(i)}}{{Im}\left\lbrack {{\hat{\alpha}}_{0}(i)} \right\rbrack}}$ where C_(II)(0)=1, C_(IQ)(0)=0, C_(QI)(0=0, C_(QQ)(0)=1 and {circumflex over (α)}(f) and {circumflex over (α)}₀(f) are the i-th estimates of the residual imbalance.
 29. The method of claim 27, wherein the compensation parameters are determined iteratively, from a starting estimate, such that G ^(tx) _(m)(i+1)=G ^(tx) _(m)(i)Ĝ ^(tx) _(m) and θ^(tx) _(m)(i+1)=θ^(tx) _(m)(f)+{circumflex over (θ)}^(tx) _(m), where Ĝ^(tx) _(m)(f) and {circumflex over (θ)}^(tx) _(m)(f) are i-th estimates of residual gain and phase imbalance.
 30. A method of compensating for transmitter and receiver imbalance in a multi-carrier, OFDM symbol transmission system, including: estimating combined transmitter and receiver gain and phase imbalance across multiple carrier channels wherein estimating includes deducing constellation points transmitted from a received signal, and calculating one or more differences between the received signal and an intended signal corresponding to the deduced constellation points; and determining compensation parameters from the combined imbalance; and loading the compensation parameters in the receiver to be used to compensate for the combined imbalance.
 31. The method of claim 30, wherein the estimating step is a frequency domain estimation, such that: ${{\overset{\sim}{a}}_{k} = {{C_{1}{\hat{a}}_{k,{eq}}} + {C_{2}{\hat{a}}_{{N - k},{eq}}^{*}}}},{C_{1} = \frac{1 + {\rho_{k}ɛ_{k}}}{1 - {ɛ_{k}ɛ_{N - k}^{*}}}}$ and $C_{2} = {- \;\frac{ɛ_{k}\left( {1 + {\rho_{N - k}^{*}ɛ_{N - k}^{*}}} \right)}{1 - {ɛ_{k}ɛ_{N - k}^{*}}}}$ where $ɛ_{k}^{({RX})} = {\frac{{\hat{a}}_{k,{eq}}^{({RX})} - a_{k}}{a_{N - k}^{*} - {{\hat{a}}_{k,{eq}}^{({RX})}\rho_{k}}}.}$ â^((RX)) _(k,eq) â_(k) is a received value of a constellation point after processing through an equalizer, k being the carrier index among the plurality of carriers; a_(k) is a transmitted value of the constellation point for a kth carrier; a^(*) _(N−k) is a complex conjugate of the transmitted value of the constellation point for an N−k th carrier, where N is a total number of carriers; and ρ_(k) is a ratio ρ_(k)=a^(*) _(N−k)/a_(k) of transmitted symbols used for equalizer training.
 32. The method of claim 30, wherein the compensation parameters are determined iteratively, from a starting estimate, such that: ${C_{1}\left( {i + 1} \right)} = \frac{{{C_{1}(i)}\left( {1 + {\rho_{k}{{\hat{ɛ}}_{k}(i)}}} \right)} - {{C_{2}^{*}(i)}{ɛ_{k}(i)}\left( {1 + {\rho_{N - k}^{*}{{\hat{ɛ}}_{N - k}^{*}(i)}}} \right)}}{1 - {{{\hat{ɛ}}_{k}(i)}{{\hat{ɛ}}_{N - k}^{*}(i)}}}$ ${C_{2}\left( {i + 1} \right)} = \frac{{{C_{2}(i)}\left( {1 + {\rho_{k}{{\hat{ɛ}}_{k}(i)}}} \right)} - {{C_{1}^{*}(i)}{ɛ_{k}(i)}\left( {1 + {\rho_{N - k}^{*}{{\hat{ɛ}}_{N - k}^{*}(i)}}} \right)}}{1 - {{{\hat{ɛ}}_{k}(i)}{{\hat{ɛ}}_{N - k}^{*}(i)}}}$ where C₁(0)=1, C₂(0)=0 and {circumflex over (ε)}_(k)(f) is the i-th estimate of a combined imbalance for the k-th subcarrier.
 33. The method of claim 30, wherein a receiver component of the combined imbalance has a predetermined value, the compensation parameters for the receiver component of the combined imbalance are time domain parameters and the compensation parameters for transmitter component of the combined imbalance are frequency domain parameters.
 34. The method of claim 33, wherein the estimating step is a frequency domain estimation, such that: {tilde under (a)} _(k) =C ₁ â _(k,eq) +C ₂ â ^(*) _(N−k,eq), where ${C_{1}\left( {i + 1} \right)} = {{{C_{1}(i)}\;\frac{\left( {1 + \rho_{k}} \right) + {{\hat{\beta}(i)}\left( {1 - \rho_{k}} \right)}}{2\left( {{\hat{\beta}(i)} + {{\hat{\beta}}^{*}(i)}} \right)}\left( {1 + {{\hat{\beta}}^{*}(i)}} \right)} - {{C_{2}^{*}(i)}\;\frac{\left( {1 + \rho_{N - k}^{*}} \right) + {{\hat{\beta}(i)}\left( {1 - \rho_{N - k}^{*}} \right)}}{2\left( {{\hat{\beta}(i)} + {{\hat{\beta}}^{*}(i)}} \right)}\left( {1 - {\hat{\beta}(i)}} \right)}}$ ${C_{2}\left( {i + 1} \right)} = {{{C_{2}(i)}\;\frac{\left( {1 + \rho_{k}} \right) + {{\hat{\beta}(i)}\left( {1 - \rho_{k}} \right)}}{2\left( {{{\hat{\beta}}^{*}(i)} + {\hat{\beta}(i)}} \right)}\left( {1 + {{\hat{\beta}}^{*}(i)}} \right)} - {{C_{1}^{*}(i)}\;\frac{\left( {1 + \rho_{N - k}^{*}} \right) + {{{\hat{\beta}}^{*}(i)}\left( {1 - \rho_{N - k}^{*}} \right)}}{2\left( {{\beta^{*}(i)} + {\hat{\beta}(i)}} \right)}\left( {1 - {\hat{\beta}(i)}} \right)}}$ where C₁(0)=1, C₂(0)=0 and β(f) is the f-th estimate of a residual far-end TX imbalance.
 35. The method of claim 34, wherein i takes a single value, i=0, subject to C₁(0)=1,C₂(0)=0.
 36. The method of claim 33, wherein the estimating step is a frequency domain estimation, such that {tilde under (a)} _(k) =C ₁ â _(k,eq) +C ₂ â ^(*) _(N−k,eq), where ${C_{1}\left( {i + 1} \right)} = {{{C_{1}(i)}\;\frac{1 + {\hat{\beta}(i)}}{2\left( {{{\hat{\beta}}^{*}(i)} + {\hat{\beta}(i)}} \right)}\left( {1 + {{\hat{\beta}}^{*}(i)}} \right)} - {{C_{2}^{*}(i)}\;\frac{1 + {{\hat{\beta}}^{*}(i)}}{2\left( {{\beta^{*}(i)} + {\hat{\beta}(i)}} \right)}\left( {1 - {{\hat{\beta}}^{*}(i)}} \right)}}$ ${C_{2}\left( {i + 1} \right)} = {{{C_{2}(i)}\;\frac{1 + {\hat{\beta}(i)}}{2\left( {{{\hat{\beta}}^{*}(i)} + {\hat{\beta}(i)}} \right)}\left( {1 + {{\hat{\beta}}^{*}(i)}} \right)} - {{C_{1}^{*}(i)}\;\frac{1 + {{\hat{\beta}}^{*}(i)}}{2\left( {{\beta^{*}(i)} + {\hat{\beta}(i)}} \right)}\left( {1 - {\hat{\beta}(i)}} \right)}}$ where C₁(0)=1, C₂(0)=0 and {circumflex over (β)}(i) is the i-th estimate of a residual far-end TX imbalance.
 37. The method of claim 36, wherein i takes a single value, i=0 subject to C₁(0)=1, C₂(0)=0.
 38. The method of claim 30, wherein the estimating step is a frequency domain estimation performed after processing a received signal through an equalizer, such that: a metric ε_(k)ε^(*) _(N−k)|_(TX only) is substantially minimized. ${{ɛ_{k}ɛ_{N - k}^{*}}}_{{TZ}\mspace{14mu}{only}} = \frac{1 - {2G_{m}^{tx}\cos\;\theta_{m}^{tx}} + {G_{m}^{tx}G_{m}^{tx}}}{1 + {2G_{m}^{tx}\cos\;\theta_{m}^{tx}} + {G_{m}^{tx}G_{m}^{tx}}}$ and ${ɛ_{k} = \frac{{\hat{a}}_{k,{eq}} - a_{k}}{a_{N - k}^{*} - {{\hat{a}}_{k,{eq}}\rho_{k}}}},$ where â_(k) is a received value of a constellation point, k being the carrier index among the plurality of carriers; a_(k) is a transmitted value of the constellation point for a kth carrier; a^(*) _(N−k) is a complex conjugate of the transmitted value of the constellation point for an N−k th carrier, where N is a total number of carriers; G^(rx) _(m) is gain imbalance of a receiver's mixer; θ^(rx) _(m) is phase imbalance of the receiver's mixer; and ρ_(k) is a ratio ρ_(k) =a ^(*) _(N−k) /a _(k) of transmitted symbols used for equalizer training.
 39. A system of compensating for transmitter imbalance in a multi-carrier, OFDM symbol transmission system, including: a receiver including logic to estimate transmitter gain and phase imbalance in a frequency domain across multiple carrier channels and to determine compensation parameters from imbalances, wherein estimating transmitter gain and phase imbalance in a frequency domain across multiple carrier channels and determining compensation parameters from imbalances includes deducing constellation points transmitted from a received signal; and calculating one or more differences between the received signal and an intended signal corresponding to the deduced constellation points; and a transmitter having memory to store the compensation parameters, whereby the compensation parameters can be used to compensate for transmitter gain and phase imbalance; and a transmitter parameter loader, coupled to the logic to estimate and to the memory to store the compensation parameter.
 40. A system of compensating for transmitter imbalance is a multi-carrier, OFDM symbol transmission system, including: a transmitter having a frequency domain analyzer section including logic to estimate transmitter gain and phase imbalance across multiple carrier channels from a transmission signal and to determine compensation parameters from the imbalances, wherein estimating transmitter gain and phase imbalance in a frequency domain across multiple carrier channels and determining compensation parameters from imbalances includes deducing constellation points transmitted from a received signal; and calculating one or more differences between the received signal and an intended signal corresponding to the deduced constellation points; and memory to store the compensation parameters, coupled to the transmitter, whereby the compensation parameters can be used to compensate for transmitter gain and phase imbalance; and logic to load the compensation parameters into memory.
 41. A system of compensating for receiver imbalance in a multi-carrier, OFDM symbol transmission system, including: a receiver including logic to estimate receiver gain and phase imbalance in a frequency domain across multiple carrier channels and to determine compensation parameters from the imbalances, wherein estimating receiver gain and phase imbalance in a frequency domain across multiple carrier channels and determining compensation parameters from the imbalances includes deducing constellation points transmitted from a received signal, and calculating one or more differences between the received signal and an intended signal corresponding to the deduced constellation points; and memory to store the compensation parameters coupled to the receiver, whereby the compensation parameters can be used to compensate for the imbalances; a receiver parameter loader, coupled to the logic to estimate and to memory to store the compensation parameters.
 42. A system for compensating for transmitter and receiver imbalance in a multi-carrier, OFDM system, including: a receiver including logic to estimate combined receiver and transmitter gain and phase imbalance in a frequency domain across multiple carrier channels and to determine compensation parameters from the imbalances, wherein estimating combined receiver and transmitter gain and phase imbalance in a frequency domain across multiple carrier channels and determining compensation parameters from the imbalances includes deducing constellation points transmitted from a received signal, and calculating one or more differences between the received signal and an intended signal corresponding to the deduced constellation points; and memory to store the compensation parameters, coupled to the receiver, whereby the compensation parameters can be used to compensate for the imbalances; and a receiver parameter loader, coupled to the logic to estimate and to the memory to store the compensation parameters. 